#### 3.2. Detachment-Limited/Transport-Limited Issue

[22] In contrast, if *ξ* is small, the sediment load is everywhere close or equal to its equilibrium value (assuming that sediment supply is large enough) *q*_{s} = *q*_{s}^{eq} (which is physically consistent with the fact that *ξ* is the distance to reach equilibrium). The mass balance equation then becomes

where **u** is the flow direction. To demonstrate this, we may imagine the mass balance applied to an elementary along-stream pixel of coordinates [*x*, *x* + Δ*x*]. The topographic increase comes from sediments that are eroded upstream and deposited within the pixel. Since *ξ* is the travel distance of sediments, only sediments located on average at a distance less than *ξ* from the upstream boundary are going to deposit within the pixel. A similar reasoning is applied to sediments that come out of the pixel, which leads to an expression of the net topographic variation

equivalent to equation (11) in the limit when Δ*x* approaches 0. Equation (11) is a typical expression for the so-called transport-limited (TL) equation, in which the sediment flux *q*_{s}^{eq} = is

In a model where *ξ* is taken constant, the slope-area relationship exhibits detachment-limited behavior at short distance and transport-limited at large distance [*Whipple and Tucker*, 2002], with a transition area proportional to the square root of the drainage area *A* (actually the transition occurs when = *ξ*, with *W* the river width).

[23] In the *ξ*-*q* model, *ξ* is small for small drainage areas and large for large basins. But it does not mean that the system goes from transport-limited regime on the drainage divide to detachment-limited regime at river mouth since it depends on the reference taken to define the notion of small and large. This aspect is discussed below.

#### 3.4. Dynamic Implications: Steady State and the Slope-Area Relationship

[25] We investigate the steady state topography that results from the *ξ*-*q* model. Steady state is defined as a dynamic equilibrium, for which both erosion and deposition rates keep pace with uplift rates. The slope-area relationship at any point *P* of topography is derived from the mass balance equations (3) by considering that both topography *h* and unit river outflow *q*_{S} are stationary

At equilibrium, *q*_{S} is the total upstream eroded material (1 − ϕ)*AT* divided by the river width *W*, where *A* is the drainage area of the basin whose outlet is *P*

The unit width discharge is likewise related to the upstream rainfall rate *r* that effectively contributed to discharge (also called effective rainfall rate thereafter)

entailing that the ratio = . Equation (13) then writes

By replacing by a power law equation (5), we can derive the basic equation for the slope-area relationship

The eventual slope-area relationship is derived by replacing *q* by the drainage area (by using the empirical equation that links the river width with water discharge, generally *W* ∼ *Q*^{0.5} if we neglect the potential dependency of *W* on incision rate, sediment supply and rock lithology [*Ferguson and Church*, 2004; *Lave and Avouac*, 2001; *Turowski et al.*, 2007].

[27] 1. If the net settling velocity *v*_{s} is independent of *q* and *s*, the deposition term in the right-hand term of equation (15) is independent of the drainage area. Thus the slope-area relationship exhibits a single scaling whatever drainage area, with an exponent close to −*m*/2*n*. The form of this equation is similar to that of the detachment-limited case. This result contrasts with previous theories that include both detachment-limited and transport-limited processes, which predict two scaling relationships [*Whipple and Tucker*, 2002] at short (detachment-limited) and large distances (transport-limited). Note that these two scaling relationships, although theoretically predicted, has never been justified by data.

[28] 2. Although the disequilibrium length increases with drainage area, this does not correspond formally to a downstream transition from a transport-limited to detachment-limited regime. Because *q*_{s} and *ξ* are both proportional to *A*/*W*, the deposition rate (equation (8)) is constant along stream, and thus so is the erosion rate, which is equal at steady state to *T*(1 − ϕ) − according to equation (3). The ratio of deposition over erosion, which is a good indicator of the detachment or transport-limited character of the dynamics, is thus constant along stream.

[29] Equation (15) can be compared with similar expression obtained for the detachment-limited and transport-limited models (equation (12) is taken for the transport-limited model)

Transport-limited

All the three erosion/deposition models end up with the same form of slope-area relationships.

#### 3.5. Dynamic Implications: Low-Frequency Evolutions

[30] It is beyond the scope of this paper to explore all the consequences of this erosion/deposition model, in particular when adding some complexity in the erosion term (tool and cover effects, erodability difference along stream between bedrock and alluvium). We just discuss simple cases that can be used as benchmark of erosion laws. The first case is the time required to reach equilibrium for a plateau submitted to a vertical uplift. The erosion/deposition model was compared to both detachment-limited and transport-limited processes defined by equations (10) and (11), respectively. The calculation was performed with the following hypothesis and equations.

[31] The fluvial erosion/transport process is modeled by using equations (3) and (4) for both mass balances. Variables are given in a dimensionless form defined such as the grid mesh, the uplift rate *T*, and the effective rainfall rate *r* are all equal to 1. The river width is supposed to vary as the square root of the total discharge, entailing that *W* ∼ *q* (for simplicity reasons, we take *W* = *q*, which leads to *q* = ).

[32] The erosion equation is a simplified version of (5) = *Kq*^{1.0}*s* = *KQ*^{0.5}*s*. The deposition model is defined by the expression of the disequilibrium length *ξ*(*q*) = *ξ*_{o}*q* = *ξ*_{o}*Q*^{0.5}. By varying *ξ*_{o}, we expect to investigate several behaviors. Note that *ξ*_{o} = 1 corresponds to a case where *ξ* is about equal to the flow path length everywhere (the demonstration is similar to that of the Hack's law).

[33] According to equations (10) and (11), both the equivalent detachment-limited and transport-limited models can be encompassed in the same mathematical framework by taking either *ξ* = ∞ for detachment-limited, or *ξ* ≪ 1 and equation (12) for transport-limited.

[35] In addition to these fluvial erosion/transport equations, hillslopes are shaped by another process that is responsible for the classical convex/concave hillslope shape. The choice of such a process is rather arbitrary but not critical for this study. Indeed, although hillslope processes control the time scale of system dynamic, it does it by a scaling factor that is independent of fluvial process [*Davy and Crave*, 2000]. Thus the results do not depend on hillslope processes in a relative sense as long as they are the same for all simulations. This assessment has been verified by testing different hillslope laws.

[36] The following hillslope process was chosen for the realism of the eventual topography: a Fickean diffusion with a diffusion coefficient *D* = 10^{−4}, associated with a linear erosion/transfer process corresponding to equation (11) with *q*_{s} = = 3.0 * *Q*^{1} * *s*^{1}. This process operates for water flow less than 10 (in pixel units).

[37] The calculations were performed for a grid of 256 × 256 with the mixed Eulerian-Lagrangian code Eros [*Crave and Davy*, 2001; *Davy and Crave*, 2000; *Lague et al.*, 2003; *Loget et al.*, 2006]. The results of the simulations are shown in Figure 1 for 3 different stages of the topography history. Note that, although the slope-area relationships are similar for all models, the eventual average altitude at steady state can slightly vary because of small differences in the fluvial network organization (Figure 1).

[38] As intuitively expected, the *ξ*-*q* model can behave either as its detachment-limited equivalent if *ξ*_{o} > 1, or as a transport-limited model if *ξ*_{o} < 1. In the former (this model *ξ* = 10*q* and detachment-limited model *ξ* = ∞ in Figure 1), most of the erosion concentrates into large rivers at the first stages, with a fast upstream propagation; steady state is reached much faster than in the detachment-limited case (dimensionless time less than 10). In the latter (this model *ξ* = 10^{−1}*q* and transport-limited model *ξ* = 10^{−1} in Figure 1), the erosion is widespread even at the first stages of process. The case *ξ* = *q* is intermediate between both end-members.

[40] Θ can be related to the ingredients of the physical model by using equations (7) and (16)

*v*_{S}, the volume average vertical velocity of particles in river, takes a large range of values, for example between 10^{−6} and 10^{−1} m s^{−1}. Here *r* is the effective rainfall, that is the ratio between discharge and drainage area. A reasonable upper bound of its annual average is about 10^{−7} m s^{−1} (corresponding to a rainy climate of 3 m per year), while values about 10^{−5} m s^{−1} (3 cm in a hour) or more are frequently encountered during the main erosive events. This back of the envelope calculation shows that the erosion mode number Θ is generally larger than 1, which is consistent with a transport-limited mode, but detachment-limited mode is likely to occur either for very small particles, or for intense climate event, or if *d** is much smaller than 1 (see the discussion thereafter).

[42] The time partitioning that we invoke for cover and bedrock erosion addresses directly the issue of flow variability whose consequences have yet to be derived in the framework of this model. Since this can lead to quite long developments, we leave this issue for future work.

#### 3.6. Dynamic Implications: High-Frequency Transient State

[43] We suspect that the response of the *ξ*-*q* model to fast perturbations (such as very fast base level drop) can be significantly different from both end-member models. We have argued in the previous paragraph that the long-term evolution of *ξ*-*q* models (model 3 in Figure 1, for instance) can be equivalent to transport-limited model. Conversely, we suspect that the large disequilibrium length *ξ* encountered downstream should produce detachment-limited-like evolutions at least during a short period.

[44] To illustrate this dual behavior, we modify the boundary conditions of the topographies eroded during the 5 experiments described in the previous section, by decreasing suddenly the altitude of the outlets of drainage basins. Before this base level drop, the topographies are at steady state. Figure 2 gives the changes of the longest stream topographic profile at different times for the model 3 (*ξ*(*q*) = 0.1*q*). It shows the inland propagation of the erosion wave subsequent to the base level drop.

[45] The altitude changes during each experiment are plotted in Figure 3 for the detachment-limited (Figure 3e), transport-limited (Figure 3f), and *ξ*-*q* cases (Figures 3a–3d) models, respectively. The curve is obtained by subtracting the stream profile at a given time and the initial profile at *t* = 0. The dotted line indicates the initial step conditions. We choose to compare the *ξ*-*q* model 3 (*ξ* = 0.1*q*) with both the detachment-limited and transport-limited equivalent models.

[46] The step boundary condition is expected to propagate inland and to smooth out because of the diffusion part of the erosion/transport equations. The relative contribution of the former (inland propagation) versus the latter (smoothing) is a signature of the erosion mode. The DL model is close to a wave propagating model, while the TL model contains both propagation and smoothing at the very first stages, but rapidly establish into a diffusion-like regime (Figure 3).

[47] As expected the *ξ*-*q* model behaves as DL for Θ < 1 (see above for a definition). On the other hand, the *ξ*-*q* model with Θ = 100 has the expected TL-like behavior. But a difference with the long-term response can be observed for the case Θ = 10 (*ξ*_{o} = 0.1), which has now some aspects of DL-like behavior for this simulation. We suspect that this discrepancy is related to the absolute transfer distance *ξ* of the river outlet, i.e., where the perturbation is applied. A DL-like propagation is defined by *ξ* larger than the distance over which the perturbation propagates. In the simulations, *ξ* is about equal to 150 *ξ*_{o} at the largest basin outlet (i.e., ). It cannot be considered as negligible compared to the stream length (∼100) if *ξ*_{o} = 0.1 (Figure 3c). This example shows up that the *ξ*-*q* model cannot be fully mapped on to the equivalent DL or TL models, and that typical features of DL dynamics (knickpoints in particular) can be reproduced without being strictly in DL.