## 1. Introduction

[2] Sediment transport in gravel bed flows is probably one of the most intriguing (and difficult) problems among a wide range of processes observed in rivers. This problem combines various physical phenomena, many of which still await clarification and better understanding. In this paper we consider one of probably the least studied sediment transport phenomena, bed particle diffusion, whose understanding is still in its infancy. Indeed, although from time to time terms “diffusion” and “dispersion” are used in research papers they often describe processes or parameters of only marginal relation to the true diffusive properties of bed particle motion. The existing knowledge and research intuition suggest that bed particle motion should be diffusive, at least on the plane parallel to the bed, with *x* axes along the flow and *y* axes across the flow. Mathematically, diffusion processes are described by the scaling growth in time of the central moments of particle coordinates, i.e., and , where *X*(*t*) and *Y*(*t*) are longitudinal (along the flow) and transverse (across the flow) coordinates of bed particles, *X*′(*t*) = *X*(*t*) − (*t*), *Y*′(*t*) = *Y*(*t*) − (*t*), *q* is the moment order, and overbar denotes ensemble averaging. For the normal (Fickian) diffusion of bed particles we have for even moments γ_{x}(*q*) ≡ γ_{y}(*q*) ≡ γ ≡ 0.5, while all odd moments are equal to zero. This diffusion regime is consistent with theoretical considerations of *Einstein* [1937, 1942] and his followers in stochastic studies of bed particle motion [e.g., *Yang and Sayre*, 1971; *Stelczer*, 1981; *Sun and Donahue*, 2000]. Indeed, it is usually assumed that probability distributions for both length steps and rest periods are exponential (or close to exponential), and thus the Central Limit Theorem applies, leading to γ_{x}(*q*) ≡ γ_{y}(*q*) ≡ γ ≡ 0.5, [e.g., *Yang and Sayre*, 1971; *Bouchaud and Georges*, 1990; *Weeks et al.*, 1996]. The normal particle diffusion is often used, implicitly or explicitly, also in experimental studies, as, for instance, that by *Drake et al.* [1988]. However, because of experimental difficulties, there have been no systematic studies, to our knowledge, convincingly confirming this assumption for a wide range of scales.

[3] In fact, there may potentially be several diffusion regimes in bed particle motion: ballistic [γ_{x}(*q*) ≡ γ_{y}(*q*) ≡ γ ≡ 1], superdiffusive [(γ_{x}, γ_{y}) > 0.5], normal [γ_{x} = γ_{y} = 0.5], or subdiffusive [(γ_{x}, γ_{y}) < 0.5]. Diffusion with γ ≠ 0.5 is known as anomalous diffusion [e.g., *Havlin and Ben-Avraham*, 1987; *Bouchaud and Georges*, 1990; *Metzler and Klafter*, 2000]. In turn, following *Castiglione et al.* [1999], the anomalous diffusion can be classified as weak [γ(*q*) ≡ γ = CONST ≠ 0.5] or strong [γ = γ(*q*) ≠ 0.5]. Also, scaling properties for anomalous diffusion may be isotropic [γ_{x}(*q*) = γ_{y}(*q*) ≠ 0.5] or anisotropic [γ_{x}(*q*) ≠ γ_{y}(*q*) ≠ 0.5]. It is important to note that the diffusion exponents may directly relate to parameters of probability distributions of particle motion characteristics such as length steps and/or rest periods [*Bouchaud and Georges*, 1990; *Weeks and Swinney*, 1998; *Weeks et al.*, 1996], as well as to statistical properties of the bed surface [*Havlin and Ben-Avraham*, 1987; *Wang*, 1994]. Sometimes diffusion properties may be different for different ranges of scales, depending on diffusion-generating mechanisms.

[4] In general, the normal diffusion (e.g., Brownian motion) results from the Central Limit Theorem of probability theory. In case of anomalous diffusion the usual form of the Central Limit Theorem fails because of the presence of either “broad” distributions (with diverging first or second moment) and/or of “long-range” correlations [*Bouchaud and Georges*, 1990]. These statistical mechanisms may be induced by external fields (e.g., long-range correlations in the velocity field of a turbulent flow may potentially generate superdiffusive motion of bed particles), or they may be induced by the dynamics itself (e.g., trapping of particles on the bed may generate “broad” distributions of rest periods leading to subdiffusion). The theoretical identification of diffusion regimes and scaling exponents for moving bed particles is difficult (if possible at all) and should be supplemented with experimental studies [e.g., *Habersack*, 2001] and computer simulations [e.g., *McEwan et al.*, 2001]. In this paper we first suggest a conceptual model of bed particle diffusion, and, second, examine field measurements which provide convincing support for this model.