## 1 Introduction

Estimating sediment transport rate in the presence of bed forms has been challenging researchers for many decades since the early work of *Bagnold* [1946]. The wide range of bed form types and geometries, as well as their dynamic interactions, has always been a challenge to a generalized statistical approach to estimating sediment transport under varying discharge and grain size distributions. Large‒scale bed forms such as alternate bars, for instance, obey specific scaling relationships and mechanisms of instability, as emphasized by a number of studies [*Ikeda*, 1984; *Colombini et al.*, 1987; *Seminara*, 2010, and references therein]. These are different from those of dunes [*Engelund and Fredsoe*, 1982; *Kennedy*, 1963; *Best*, 2005; *Venditti et al.*, 2005a; *Coleman et al.*, 2006; *Colombini and Stocchino*, 2011], ripples, chevrons, or antidunes [*Stegner and Wesfreid*, 1999; *Betat et al.*, 1999; *Fourriere et al.*, 2010; *Andreotti et al.*, 2012]. Some of these contributions employed linear stability analysis to explore how different bed form types result from the amplification of specific initial perturbations starting from flatbed conditions and how, for each bed form type, a key length scale (e.g., scaling with the channel width or depth) can be derived corresponding to the most unstable perturbation. These theoretical predictions have been validated by laboratory experiments reporting the dominant bed form wavelength and height [*Guy et al.*, 1966; *Jaeggi*, 1984; *Colombini et al.*, 1987, and references therein]. In many realistic conditions, however, different bed form types and sizes were observed to overlap, interact and/or merge into complex migrating bed forms [see, e.g., *Jerolmack and Mohrig*, 2005; *Venditti et al.*, 2005b; *Martin and Jerolmack*, 2013].

In addition, bed forms of the same type can exhibit a significant variability and thus display a wide range of wavelengths and heights [*Hino*, 1968; *Nikora et al.*, 1997; *van der Mark et al.*, 2008; *McElroy and Mohrig*, 2009; *Singh et al.*, 2011, among others]. The statistical variability in bed form shape and size and the possible superimposition of different bed form types suggest that several length and timescales need to be accounted for in capturing the evolution of complex riverbed topography in natural environments [*Allen*, 1968; *Rubin and McCulloch*, 1968]. While studying the initial growth of competing perturbations of different sizes poses several theoretical challenges, the evolution of multiscale bed topography can be addressed rather simply, once bed forms have reached a statistically steady state. We consider herein the case of migrating bed forms in a straight flume in dynamical equilibrium conditions, and we focus on their spatiotemporal evolution and their effect on sediment transport. We specifically explore a framework in which bed form classification and extraction is not strictly necessary to quantify their contribution to sediment transport. Instead of working in the physical domain to estimate individual bed form heights and lengths, we decompose bed elevation series into a space‒time wave number, frequency domain, and integrate over a range of scales to capture the contribution of the multiscale migrating features of complex bed topography to the total sediment transport.

Since the early work of *Simons et al.* [1965] the connection between sediment transport rate and bed form characteristics has been expressed based on a geometric formulation, as

where *q*_{s} is the transport rate per unit width, *p* is the porosity of the bed material, *V*_{b} and *H*_{b} are the velocity and height of the statistically dominant, or average, bed form, respectively. The factor 2 in the right‒hand side of equation (1) comes from the assumption of a triangular bed form shape, and *q*_{0} represents the contribution to bed load transport that does not enter into the propagation of bed forms. Equation (1) can also be derived from the integration of the Exner equation [*Paola and Voller*, 2005] assuming that, in equilibrium condition, bed forms do not deform while migrating; *q*_{0} appears then as an integration constant which is neglected here [see, e.g., *McElroy and Mohrig*, 2009].

Equation (1) requires only estimates of the average values of *V*_{b} and *H*_{b} which can be provided using any bed form extraction technique able to identify period and wavelength of each bed form [*van der Mark et al.*, 2008; *McElroy and Mohrig*, 2009; *Singh et al.*, 2011]. Nevertheless, the spatial (or temporal) variability in the bed form geometry, e.g., measured by the standard deviation of an ensemble of bed form heights and wavelengths extracted from the bed topography, is not incorporated in the *Simons et al.* [1965] formulation, which accounts only for the average values. In addition, the coexistence of various sizes of topographic features and types of bed forms, e.g., ripples superimposed on dunes, makes the *Simons et al.* [1965] approach too simplistic, as large bed forms are known to travel slower than smaller bed forms [*Hino*, 1968; *Nikora et al.*, 1997; *Coleman and Melville*, 1994; *Giri and Shimizu*, 2006; *Schwämmle and Herrmann*, 2004; *Singh et al.*, 2011]. This induces a second element of dynamic variability which adds to the intrinsic spatial variability of bed forms, resulting in a modulated temporal evolution of the bed topography. Bed form‒dependent propagation velocities are therefore needed to include the effects of spatiotemporal bed form variability and evolution and ultimately extend the *Simons et al.* [1965] approach to the case of complex erodible riverbed topographies with multiscale topographic features and migrating bed forms. By treating any migrating surface feature as a combination of Fourier modes, we introduce a generalized approach to calculating propagation velocities avoiding bed form classification. The propagation velocities of different size bed features are statistically described here in terms of spectral convection velocities *C*_{V}=*ω*/*k*, where *k* and *ω* are the wave number and frequency of the different scale Fourier modes in which the evolving topography is decomposed.

Similar ideas have been used in the turbulent boundary layer community where a spectral approach has been pursued to estimate scale‒dependent convection velocities to test the applicability of Taylor's hypothesis of frozen turbulence [*Morrison et al.*, 1971; *Erm and Joubert*, 1991; *Krogstad et al.*, 1998; *Dennis and Nickels*, 2008; *Chung and McKeon*, 2010; *LeHew et al.*, 2011, among others]. Essentially, Taylor assumed that turbulent flow structures on average move with the local (i.e., at fixed height) mean velocity, implying that a velocity signal recorded in time can be projected in space via a simple renormalization with no information loss. Testing Taylor's hypothesis requires that the convection velocity of different turbulent structures can be estimated. Since those structures are not easily identifiable in the physical domain, the spectral approach is pursued using velocity measurements simultaneously obtained in time and space. Time‒resolved particle image velocimetry measurements were used for this specific purpose in *LeHew et al.* [2011] in the case of a flat plate turbulent boundary layer flow. Scale‒dependent convection velocities were then estimated by means of two‒dimensional spectra in the frequency and wave number(s) domain. We propose here a similar approach to quantify the scale‒dependent velocities of migrating bed forms based on measurements of evolving surface topography *z*=*z*(*x*,*t*), instead of flow velocity spatiotemporal series *v*=*v*(*x*,*t*). Note that a quantification of the propagation velocities of bed forms of different scales was performed by *Singh et al.* [2011] using only temporal elevation data and employing a wavelet‒based correlation analysis. That study suggested that smaller bed forms move faster than larger ones.

Experimental observations of large bed forms statistically propagating slower than smaller bed forms [*Coleman and Melville*, 1994; *Giri and Shimizu*, 2006; *Schwämmle and Herrmann*, 2004; *Singh et al.*, 2011; *Martin and Jerolmack*, 2013] can be interpreted as the result of smaller bed forms merging into larger ones. This mechanism suggests that (i) the evolution of bed topography is the result of the coupling between bed form advection and deformation processes; therefore, (ii) Taylor's assumption of a single convective velocity is not expected to represent correctly migrating bed forms in nature, and ultimately, (iii) *Simons et al.* [1965] approach may not provide an accurate estimate of sediment transport rate in complex topographic conditions. These observations suggest the possibility of extending *Simons et al.* [1965] model to include multiscale bed form migration and deformation processes through the introduction of scale‒dependent convection velocities. Such a conceptual framework is shown in Figure 1, where different size bed forms are represented by different sinusoids: note that as the bed form height is conventionally defined from trough‒to‒crest, the corresponding sinusoid amplitude must be doubled and its specific sediment discharge contribution is proportional to *A*(*k*)*C*_{V}(*k*). We stress that multiscale bed forms are intended here to account for the multiscale variability of the evolving bed topography, including as special cases the coexistence of different bed form types as well as the spatiotemporal variability within each bed form type.

The first goal of this paper is to propose a spectral description of evolving surface bed elevations, in both the frequency and wave number domain, and provide a method to estimate scale‒dependent convection velocities for migrating topographic features. The second goal is to identify, for the case of migrating bed forms in equilibrium conditions, a scaling relationship between the period and wavelength of the Fourier decomposed bed forms, propose a set of normalizing variables and a dimensionless description of convection velocities. Finally, we pursue a generalization of the *Simons et al.* [1965] approach toward the formulation of a sediment transport rate model that explicitly acknowledges the presence of space‒time topographic feature of multiple scale. Measurements of evolving bed elevations, *z*=*z*(*x*,*t*), were obtained by a partially submerged sonar mounted on a computer‒controlled cart, covering the flume's central longitudinal transect of 5 m every 13–20 s (depending on the run), for a total of approximately 20 h. Two different sets of experiments were conducted under varying flow discharges using sand (set 1) and fine gravel (set 2) as bed material, in a range of Froude numbers 0.2<*F**r*<0.5.

The paper is structured as follows. The experimental setup is described in section 2. Spectral analysis is introduced in section 3.1, while scale‒dependent convection velocity and key scaling arguments are discussed in section 3.2. An expression for estimation of sediment transport rate in the presence of bed forms is proposed in section 3.3, whereas an alternative scaling option is discussed in section 4. Finally, concluding remarks are given in section 5.