[7] Up to this point, research on the application of equation (5), or its equivalent (6), in modeling landscapes has examined the role of the locality factor *α*, with a focus on understanding how this value is related to the statistics of physically observed features in the landscape [*Schumer et al.*, 2009; *Foufoula-Georgiou et al.*, 2010; *Voller and Paola*, 2010]. Here we study instead the non- locality direction coefficient −1 ≤*β* ≤ 1. In particular, on using the basic properties of Caputo derivatives [*Foufoula-Georgiou et al.*, 2010; *Podlubny*, 1998]—that the fractional derivative of a positive power is

the derivative of a constant is identically zero, and in the interval 0 ≤ *x* ≤ 1, the right and left derivatives are related through [*Foufoula-Georgiou et al.*, 2010]

—we can re-solve(1) and (2) with the unit discharge definition in (6)and arrive at non-local expressions for the fluvial profile in the limit cases of*β* = 1 or *β* = −1 (Table 1). Plots of the source-sink profiles inTable 1 (Figure 2) for a non-locality of*α*= 0.7 show that with a full upstream bias in the locality, the maximum surface elevation, as physically expected, is located at the origin. On the other hand, if we assume a full downstream bias the erosional section of the profile is non-physical, exhibiting a maximum elevation downstream of the origin. In fact, assuming that the profiles for general values of the direction parameter −1 <*β* < 1 must be enveloped by these limit solutions, we argue that a physical meaningful solution for the erosion profile, with the maximum elevation located at the origin, can occur only with a full upstream bias *β* = 1. Similar arguments lead to the conclusion that a physically meaningful depositional profile solution can occur only with a full downstream bias *β*= −1. Hence, to the extent that the fractional calculus provides a valid non-local treatment, the mathematical analysis of long-profiles solutions compared to the physically expected behavior leads to an interesting hypothesis:

[8] Non-locality in an erosional landscape is purely determined by upstream features and conditions, whereas in a depositional landscape the non-locality is controlled by downstream features and conditions.

[9] This hypothesis implies that the flow of information in landscape dynamics depends on the nature of the system considered; in an erosional system, information flows forward, in the downstream direction whereas in a depositional system, information flows backwards, in the upstream direction.