#### 4.1. Initial and Boundary Conditions

[46] The transformed equations (24) and (27) represent the governing equations for the time evolution of the bed elevation in the bed region *η*_{b} and the channel bottom width *B*_{b}, respectively. Since both of these coupled partial differential equations are highly nonlinear, they are solved numerically. One initial condition and two boundary conditions are required for *η*_{b} in equation (24), while one initial condition and only one boundary condition are needed for *B*_{b} in equation (17).

[47] The experiments performed by *Cantelli et al.* [2004] indicate that as a first reasonable approximation the reach upstream of the original dam degrades around a “pivot” point located immediately upstream of the removed dam. The numerical model does not yet explicitly include the aggradational region located downstream of the (removed) dam, though the influence of the aggradational region appears implicitly via the pivot point. Initial condition for *η*_{b} corresponds to the final longitudinal profile of the progradational delta front resulting from the previous dam-induced sedimentation.

[48] Although the initial condition for bottom width *B*_{b} can in principle be set to any value, here it is set equal to the equilibrium width associated with the specified water discharge *Q*_{w}, the slope of the delta topset *S*_{T} before incision takes place, and grain size *D*. This equilibrium width is obtained by specifying the condition that the sediment is at the threshold of motion in the sidewall region, so that _{syj} vanishes. This condition in combination with (15) yields the constraint

[50] Equations (7a) and (7b) applied to the initial configuration likewise yield the following relation for the initial width *B*_{bI}:

[51] In general this equation must be solved iteratively in combination with (7a)–(7c) and (8b), but the iteration can be commenced by setting *c*_{1} and *c*_{2} equal to unity.

[52] The reach of interest is defined to have length *L*, where *x* = *L* denotes the pivot point (just upstream of the location of the former dam) discussed above. Taking the pivot point as the elevation datum, the boundary condition there applying to takes the form

[53] A second boundary condition for *η*_{b} is derived from the assumption of a specified constant total volume bed load supply rate at the upstream end of the channel. Denoting the total volume bed load transport rate as *Q*_{x}, then,

where *Q*_{xu} denotes the supply rate of bed load. In general, *Q*_{xu} can be specified arbitrarily, but in the present work the value used is the one that is in equilibrium with the initial bed slope of the channel (such that neither aggradation nor degradation would occur if the channel were to have the initial slope in the absence of a dam).

[54] The total streamwise bed load transport rate is equal to the sum of the rates in the channel and bed regions,

but in light of the imposition of the condition that the sidewall region is at the threshold of motion, (34a) reduces to

[55] Applying (7a), (7b), (8a), (10a), (29), and (33) at the upstream end of the reach, the following boundary condition on bed slope *S*_{0} is obtained:

[56] The solution is again obtained iteratively in conjunction with (7a)–(7c) and (8b), and again the iteration can be commenced by setting *c*_{1} = *c*_{2} = 1.

[58] The terms on the left-hand side of equation (36a) take the form of a kinematic wave equation, for which disturbances propagate upstream with wave speed *c*_{B}, where

[59] The problem is, however, more complicated than this, because the term on the right-hand side of (36a) is coupled to (24). A linearized analysis of homogeneous versions of (24) and (36a) yield a second-order equation for complex celerity, suggesting the possibility of two wave speeds. The results of the numerical analysis, however, confirm the existence of an upstream propagating morphodynamic wave of the type indicated by the left-hand side of (36a).

[60] The form (25b) implies that *N*_{B} is positive for sufficiently small values of *H*/*B*_{s}, and specific calculations indicate that for most cases of interest *N*_{B} is thus positive. As a result, (36b) indicates a negative wave speed, so that the width disturbance propagates upstream. As a result, the appropriate boundary condition on (36a) should be applied at the downstream end of the reach.

[61] A reasonable assumption for this boundary condition is that the pivot point at *x* = *L* also constitutes a point of maximum bed load transport rate, so that upstream of this point the sediment load increases downstream (incising channel) and downstream of this point the sediment load decreases downstream (aggrading channel). The boundary condition on *B*_{b} can thus be obtained from the relation

Reducing (34a) with the aid of (10c) and (22), the following mixed boundary condition on *B*_{b} is found to hold at the downstream end:

**where**

[62] The term on the right-hand side of (25a) and (25b) becomes negligible for sufficiently large aspect ratio *B*_{b}/*H*. Insofar as this condition was found to prevail in the present analysis, (38a) has been approximated to the form