Reply to comment by Z. B. Wang and H. J. De Vriend on “Depth integrated modeling of suspended sediment transport”

Abstract

[1] Let us first thank Wang and De Vriend [2004] for their attention and their careful scrutiny of our paper. Let us also clarify that the motivation of our work was not to show that Galappatti's [1983] model (hereinafter referred to as G83) was incorrect, but rather to develop a depth-averaged formulation of suspended sediment transport suitable for long-term morphological predictions, able to take into account weak non equilibrium effects of the kind often encountered in tidal as well as fluvial environments. Unfortunately, the mathematical framework appropriate to perform such investigation is precisely that proposed in G83, which forced us to revisit critically the latter work. [2] Let us come to the points raised by the Wang and De Vriend, which do concern G83's approach rather than our formulation, whose validity does not seem to be questioned. The issue underlined in the discussion is whether or not G83's perturbation scheme is correct and the basis for the Wang and De Vriend's defense are the works of Wang [1992] and Wang and Ribberink [1986] (hereinafter referred to as W92 and WR86, respectively). We then also need to revisit the latter works. [6] We do not understand the rationale behind the latter procedure, which does not add much to our understanding of G83's approach and does not seem to us to be consistent. The latter statements arise from the following observations. [7] 1. Why should a weight function be introduced in equation (6)? In other words, why should we seek a solution such that its error relative to the exact solution close to the bed (where the concentration is largest) should weigh much less than close to the free surface (where the concentration is minimum)? Indeed, this is what is imposed by setting w = 1/a0 as proposed by W92. [8] 2. Stating that the classical solution is a particular case of the latter approach corresponding to the choice wa0 equal to the Dirac distribution δ(ζ) is true but obvious: with the latter choice J reduces to the square of the difference between the perturbation solution and the exact solution at the bed. This difference must obviously vanish in the correct solution. [9] 3. W92 ignores the dependence of Ψj (j > 0) on f0: why? Applying equation (7a) rather than equation (7b), we have found an inconsistency arising already at O(δ): in fact, the integral of the O(δ) component of the solution does no longer vanish! [10] Wang and De Vriend are totally right in pointing out that there is a numerical error in our original Figures 2 and 3 plotting our correct analytical results for the simple model problem (the corrected plots, Figures 1 and 2, are given below): it is then true that if one expands in G83's approach applied to such model problem, G83's expansion is more accurate than the classical expansion truncated at O(δ) but less accurate than the classical expansion truncated at O(δ2). However, Wang and De Vriend are not right when they generalize the latter statement: in fact, as extensively discussed in the paper, the accuracy of G83's solution is not known a priori. Moreover, the convergence properties of the simple model equation considered by Wang [1992, sect. 4] cannot be extended to the general case. In Figure 3 we show a comparison between the full 3-D solution, G83's solution and our solution in the case of flow on a slowly varying sinusoidal bed, having assumed the distribution of concentration to be periodic: G83's solution is slightly less accurate than the classical expansion even at O(δ). [11] Here, in our opinion, Wang and De Vriend fail to recognize the main point of our paper. [12] In fact, in the classical perturbation scheme the local concentration c is expanded in powers of δ both in the advection-diffusion equation and in the boundary conditions. At the leading order one then recovers the classical Rouse solution (i.e., the equilibrium solution) evaluated locally and instantaneously. No boundary condition is needed in this case: the local hydrodynamic conditions determine the sediment concentration and the flux of suspended load. At O(δ) corrections are found for the latter quantities which are expressed in terms of spatial (and temporal) derivatives of the leading-order solution. Let us stress that the classical approach cannot fit any boundary or initial condition as it does not lead to a differential equation but rather to an analytic solution for the depth-averaged concentration. This is reasonable inasmuch as the flow is slowly varying in space and time, hence the sediment concentration must be weakly perturbed with respect to the local and instantaneous equilibrium concentration: if such condition is not satisfied (i.e., δ is not sufficiently small) the expansion does not converge and the whole approach is invalid [see Vignoli et al., 2003]. [14] It is surprising that Wang and De Vriend dispute this criticism as Wang and Ribberink [1986] themselves have essentially proved our statement. In fact, the theoretical analysis of Wang and Ribberink [1986, sect. 3, p. 57] shows precisely that if one applies G83's approach to a transient process starting from an initial or boundary condition for the concentration which is far from equilibrium, one finds that G83's solution does not converge to the exact solution. How can one be surprised? WR86 were trying to describe a fast transient, where displays O(1) variations in a reach of length O(δ), using an approach which is only valid under slowly varying conditions. [15] It is also not surprising that in WR86, G83's solution is found to converge to the first exponentially decaying mode of the exact solution. The latter is also the slowest component of the exact solution which prevails at large distance from the entrance where the concentration distribution has attained a fairly slowly varying behavior. This again confirms that one cannot treat (8) as a differential equation being unable to consistently fit initial/boundary conditions. [16] None of Wang and De Vriend's conclusions seem appropriate to us, nor do they agree with the conclusions of WR86. Let us summarize what one of the discussers (Wang) writes in 1986: “For steady uniform flow conditions…the asymptotic solution on which Galappatti's model is based does not converge to the complete exact solution…For this reason the model is only valid for gradually varying flow…” This is precisely what we have claimed. This notwithstanding, assuming δ finite, equation (8) is ultimately a variant of the usual form of the depth-averaged advection diffusion equation. It is then not surprising that it may give reasonable results even under conditions which are not strictly slowly varying: this simply suggests that the depth-averaged approach is not so sensitive to the choice of the structure functions employed to perform the averaging.