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• W2009263229 abstract "[1] In their paper, Bolla Pittaluga and Seminara [2003] claim that Galappatti's asymptotic model concept [Galappatti, 1983] is formally incorrect. To us, understanding this conclusion is based on three arguments: (1) Galappatti's approach differs from the classical perturbation expansion method. (2) A comparison between the different approaches shows that Galappatti's solution is less accurate than the classical method. (3) Galappatti's approach has a dilemma concerning the initial and boundary conditions. In the following it is explained why we disagree with these arguments and therefore dispute the conclusion drawn by Bolla Pittaluga and Seminara. [2] That Galappatti's approach differs from the classical perturbation expansion method was already pointed out by Wang [1989, 1992], who, in fact, also worked out the classical method as presented by Bolla Pittaluga and Seminara in their paper. More importantly, Wang [1992] also pointed out that Galappatti's approach, as well as the classical method, can be considered as examples of a generalization of the perturbation expansion method. The difference between the two approaches is the profile function used for the basic differential operator determining the consecutive terms of the expansion. The classical method chooses the Dirac function at the bed and Galappatti chooses the unit function over the whole water column. This in itself does not justify claiming one of these two approaches better than the other, they are just different. [3] Wang [1989, 1992] made a comparison between the different approaches for various cases and came to the conclusion that in all these cases Galappatti's solution performs better than the classical one. We are pleased to notice that the Bolla Pittaluga and Seminara now contribute another case (equations (1a), (1b), and (1c) of Bolla Pittaluga and Seminara), but fail to see why in this case Galappatti's solution should perform less than the classical solution. Using the equations presented by Bolla Pittaluga and Seminara, we obtain different numerical results than those presented in Bolla Pittaluga and Seminara's Figures 2 and 3. In contrast to their conclusion, we conclude that the error of Galappatti's solution at the bed boundary is always zero (as expected), and that this solution, as compared to the classical one, has a smaller error in the same order of approximation (see Figures 1 and 2). The latter agrees with the conclusions drawn by Wang [1989, 1992]. The advantage of Galappatti's solution is even more evident at larger values of δ, as can be shown by numerical comparison (see Figure 3). The reasons for this are the following: (1) The expansion of Galappatti has a larger convergence domain than the classical expansion. For the test case presented by Bolla Pittaluga and Seminara, δ < ≈ 9.87 versus δ < ≈ 3.62, b being the smallest positive root of tan b = −2b. (2) In contrast to the classical solution, Gallappati's gives reasonable estimates even outside the convergence domain [Wang, 1992] (see also Figure 3). This robustness is essential to the applicability in numerical models of practical situations. [4] In the original 2DV/3D problem, the sediment concentration in the whole model domain is required as an initial condition. As shown by Wang [1992] for a simple case, an initial vertical concentration profile needs to be projected onto a set of eigenfunctions for determining the analytical solution. The full exact solution for the depth-averaged concentration contains an infinite series of decaying exponential functions of time, expressing the influence of the initial condition. Exactly one of these exponential functions is within the convergence domain of the expansion of Galappatti. Therefore only one initial condition (and one upstream boundary condition) can be applied in the model of Galappatti, even if a higher-order solution is used [Wang, 1989, 1992; Wang and Ribberink, 1986]. No initial condition, however, nor any upstream boundary condition, can be applied in the classical method, because all the exponential functions are outside the convergence domain. [5] Both methods therefore have a problem if the initial and boundary conditions are of significant influence, simply because the near-field problem is outside the convergence domain [Wang, 1992]. The difference between the two methods is that the convergence domain of Galappatti's solution is larger. Hence at least part of the influence of the initial and boundary conditions can be taken into account. [6] In summary, we conclude the following: (1) both the approach of Galappatti and the classical perturbation expansion can be considered as special cases of a generalized perturbation expansion. (2) Galappatti's solution is more accurate than the classical solution, also for the test case presented by Bolla Pittaluga and Seminara. (3) Galappatti's solution is practically more robust than the classical one because of its lager convergence domain and its behavior outside that domain. (4) the requirement of initial and upstream boundary conditions for the depth-averaged sediment concentration equation complies with the original problem, also in this aspect Galappatti's solution is more robust than the classical one. [7] The work of Galappatti [1983] [see also Galappatti and Vreugdenhil, 1985] can therefore not be claimed to be incorrect and has rightfully been embedded in various commercial codes." @default.
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• W2009263229 date "2004-10-01" @default.
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• W2009263229 title "Comment on “Depth-integrated modeling of suspended sediment transport” by M. Bolla Pittaluga and G. Seminara" @default.
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• W2009263229 doi "https://doi.org/10.1029/2003wr002620" @default.
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