## 1 Introduction

[2] There is a wide range of problems in river morphodynamics and landscape evolution where simple relationships between averaged sediment flux and averaged drivers, such as topography, sediment properties, and water discharge, provide useful predictive models, especially over large space and/or time scales. Because the flow is fundamentally gravity driven, relations of this kind are often cast in a form in which the flux is a function of the topographic slope. Although there have been numerous analyses proposing different ways to relate flux to topographic slope [*Paola*, 2000], until recently, there was little question that the slope to use in such models was the one at the point of interest. This view, however, has been challenged in a series of recent papers [e.g., *Bradley et al*., 2010; *Foufoula-Georgiou et al*., 2010; *Ganti et al*., 2010; 2011; *Ganti*, 2012; *Schumer et al*., 2009; *Stark et al*., 2009; *Voller and Paola*, 2010; *Voller et al*., 2012], which propose that in some cases the sediment flux at a point might also depend on values of the slope (or other topographic measures) away from that point. This dependence, typically expressed via some form of convolution integral, i.e., weighted average over space and/or time, takes into account the fact that probabilistic sediment motion may span a wide range of transport length scales. One end-member case is when the sediment motion exhibits a probability distribution with a power-law decaying tail (“heavy tail”). In this case, it is not possible to assign a characteristic length scale of transport [e.g., *Benson*, 1998] and consequently rigorous descriptions of macroscale morphologic evolution are best cast in terms of the so-called fractional calculus [e.g., *Podlubny*, 1999; *Foufoula-Georgiou et al*., 2010; *Furbish and Haff*, 2010; *Ganti et al*., 2010; *Ganti*, 2012; *Schumer et al*., 2009; *Voller and Paola*, 2010].

[3] The idea that system evolution at a point also depends on conditions away—potentially quite far away—from that point is referred to as “nonlocality”. Conceptually, it represents a profound shift in thinking about how influence and information are distributed and propagated in landscapes. So one might well ask: Exactly what difference does nonlocality make to overall morphologic evolution? Clearly, it is easy to see how a model using a nonlocal flux formulation could produce a much wider range of behavior outcomes (e.g., anomalous diffusion [*Chaves*, 1998; *Metzler and Klafter*, 2000]) when compared to a model that uses a standard linear diffusion treatment in which the flux is proportional to the local slope. It has been observed, however [*Foufoula-Georgiou et al*., 2010; *Voller and Paola*, 2010], that more general, but still local, flux models based on a nonlinear function of the local slope also allow for prediction of a wide set of behaviors, many of which partially or completely replicate the macroscopic behaviors arising from nonlocality. If this is so, then do we really need nonlocality?

[4] One answer is that if the underlying transport process is really nonlocal, representing the effects of that process via a contrived nonlinearity is likely to be prone to error, continual tuning, and scale dependence of the tuned parameters [e.g., *Ganti et al*., 2012]. More fundamentally, *any* local model implies that two points with the same local slope, water discharge, etc. would yield the same sediment flux, no matter where these points sit in the landscape relative to their upstream and downstream conditions. Is this really reasonable?

[5] As discussed earlier, nonlocal transport formulations are often associated with the absence of a characteristic scale of transport as manifested, for example, in power-law probability distributions of sediment travel distances (often truncated at the scale of the system). One sufficient but not necessary condition for such power-law distributions is the presence of a “transport network” or collection of pathways that exhibits fractality or internal self-similarity, e.g., the self-similar geometry of drainage networks [*Rinaldo and Rodriguez-Iturbe*, 1996], braided rivers [e.g., *Sapozhnikov and Foufoula-Georgiou*, 1996; 1999; *Foufoula-Georgiou and Sapozhnikov*, 1998; 2001; *Sapozhnikov et al*., 1998], and delta distributaries [*Wolinsky et al*., 2010; *Edmonds et al*., 2011]. If, to first order, we think of fluvial systems as comprising a network of efficient sediment transport pathways (channels and channel segments) and resting places (e.g., floodplains and bars), then the fractal geometry of the channel network system suggests a broad range of transport speeds, across multiple space and time scales. Thus, fractal geometry of the transport system leads naturally to a power-law distribution of transport steps. The fractal geometry of channels in a braided river system, for example, should result in sediment motions that span a wide range of spatiotemporal scales, and thus themselves exhibit heavy-tailed distributions. The occurrence of such heavy-tailed motions is a fundamental requirement for nonlocal transport [e.g., *Ganti*, 2012; *Ganti et al*., 2010; *Schumer et al*., 2009], raising the possibility that nonlocality could be a common, intrinsic feature of channelized transport systems.

[6] The potential influence of nonlocal transport on understanding the workings of channelized systems goes beyond simply describing the evolution of topographic profiles. For example, *Voller et al*. [2012] recently showed that nonlocality leads directly to a fundamental consequence for how topographic information propagates in landscapes: purely downstream in erosional systems and purely upstream in depositional systems. We stress that this is a statement only about how influence flows in space; the particles themselves, of course, move downstream. The applicability of insights like this depends on knowing to what extent nonlocal as opposed to nonlinear effects govern transport behavior in landscapes.

[7] To this end, untangling the effects of nonlocal versus nonlinear sediment transport dynamics has been difficult because the two approaches have been mutually exclusive. Our primary aim here is to bridge this gap by developing a combined nonlocal, nonlinear (NLNL) framework for the topographic evolution of fluvial systems. Our study highlights the important distinctions and similarities between the topographic signatures of nonlinear and nonlocal sediment transport dynamics, and we hope that it will provide insight and further impetus for more precise experiments that will allow us to explore unambiguously the presence and the relative role of nonlinearity and nonlocality in depositional systems.