Closure

Journal of Hydrology 216 (1999) 257–258

Closure D. A. Barry a,*, K. Bajracharya b b

a School of Civil and Environmental Engineering, The University of Edinburgh, Edinburgh, EH9 3JN UK Resource Sciences Centre, Department of Natural Resources, Block C, Gate 2, 80 Meiers Road, Indooroopilly, Queensland 4068 Australia

We appreciate the contributions of the discussers towards our paper. In particular, we value the points Cunge mentions concerning the applicability of the Muskingum–Cunge (MC) method, and its role in real-time flood forecasting. We would like to add to Cunge’s discussion, however, in order to clarify and amplify some issues. 1. Cunge’s description the MC method as ‘‘an approximation of an initial value problem for the diffusive wave equation’’ is at variance with typical usage of the description ‘‘initial value problem’’, which usually refers to solutions of (systems of) first-order ordinary differential equations. As we stated in our paper (Bajracharya and Barry, 1997), the MC method solves the diffusive wave equation (Eqs. (1)) (notation is unchanged):

2Q 22 Q 2Q ˆD 2 ⫺c ; 2t 2x 2x

…1†

subject to: Q…x; 0† ˆ Qi …x†;

…2†

Q…0; t† ˆ Qb …t†

…3†

* Corresponding author. Tel.: ⫹ 44 (131) 650-7204; fax: ⫹ 44 (131) 650-7276. E-mail addresses: [email protected] (D.A. Barry), Kiran. [email protected] (K. Bajracharya)

and

2Q 兩 ˆ 0: 2x x!∞

…4†

Since both initial and boundary values are specified in Eqs. (2) and Eqs. (3), a more apt description of Eqs. (1)–(4) is an initial-boundary value problem. We reiterate that, insofar as the boundary conditions Eqs. (2) and Eqs. (3) are concerned, the discretisation used in the solution must be small enough to capture the variability of the functions Qi(x) and Qb(t). 2. Cunge states that the semi-infinite domain solved by the MC method limits its application to ‘‘relatively steep rivers’’. In other work, we, and others, have examined closely the effect of imposing a downstream boundary condition at a finite distance, for which Eq. (4) is an approximation (Parlange and Starr, 1975, Parlange and Starr, 1978; Parlange et al., 1985; Parlange et al., 1992; Bajracharya and Barry, 1994). In particular, suppose the reach length under consideration is L. Then, if cL/D ⬍ 4 then the MC method should not be used (in all our examples the condition cL/D ⬎ 4 was satisfied). Otherwise, the MC solution can be corrected (Bajracharya and Barry, 1994). Moreover, on dimensional grounds, it can be seen that the influence of the downstream boundary condition is felt a distance of order D=cexp‰c…x ⫺ L†=DŠ upstream from L. Thus, the influence of the downstream condition can be estimated. Finally, for a

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D.A. Barry, K. Bajracharya / Journal of Hydrology 216 (1999) 257–258

series of reaches, Barry and Parker (1987) give conditions under which each reach can be treated as being semi-infinite without significantly affecting the results. 3. Cunge states that the MC method allows selection of the ‘‘coefficients K and X to define the diffusion coefficient’’. Certainly, this is the case. However, mathematically at least, there are an infinite number of ways in which this can be done. So, how is one to choose? One approach is to curvefit. In our paper we showed that these parameters can be chosen optimally, given knowledge of the coefficients, D and c. 4. Our application of the MC method to the River Wye followed a previous successful application of the same technique to the same river. Thus, we did not consider it necessary to re-justify its application.

References Bajracharya, K., Barry, D.A., 1994. Note on common mixing cell models. J. Hydrol. 153, 189–214. Bajracharya, K., Barry, D.A., 1997. Accuracy criteria for linearised diffusion wave flood routing. J. Hydrol. 195, 200–217. Barry, D.A., Parker, J.C., 1987. Approximations to solute transport in porous media with flow transverse to layering. Transp. Porous Media 2, 65–82. Parlange, J.-Y., Barry, D.A., Starr, J.L., 1985. Comments on ‘‘Boundary conditions for displacement experiments through short laboratory soil columns’’. Soil Sci. Soc. Am. J. 49, 1325. Parlange, J.-Y., Starr, J.L., 1975. Linear dispersion in finite columns. Soil Sci. Soc. Am. Proc. 39, 817–819. Parlange, J.-Y., Starr, J.L., 1978. Dispersion in soil columns: Effect of boundary conditions and irreversible reactions. Soil Sci. Soc. Am. Proc. 42, 15–18. Parlange, J.-Y., Starr, J.L., van Genuchten, M. Th., Barry, D. A., Parker, J.C., 1992. Exit condition for miscible displacement experiments in finite columns. Soil Sci. 153, 165–171.