1. INTRODUCTION The MuskingumCunge method of flood routing is well established in the hydrologic engineering literature (Cunge, 1969; Ponce and Yevjevich, 1978; U.S. Army Corps of Engineers, 1990). It is a convenient method because the routing parameters are a function of channel properties and grid specification, which leads to results which are independent of grid size. The method has linear and nonlinear modes. In the linear mode, average flow values are used to calculate the routing parameters at the start of the computation, and these are kept constant throughout the computation in time. In the nonlinear mode, the routing parameters are recalculated for every computational cell as a function of local flow values. Both modes of computation have their advantages and disadvantages. The linear mode conserves mass, but it requires a priori estimation of a reference flow on which to base the computation of the routing parameters. In addition, the routed flows lack the steepening exhibited by inbank flood waves. Conversely, the nonlinear mode does not require the specification of a reference flow and can simulate the wave steepening. Unfortunately, the nonlinear mode is saddled with a small but perceptible loss of mass. Given the tradeoffs involved, it seems certain that both routing modes will continue to be used in the future. The linear mode will be used where simplicity is desired, notwithstanding its lack of wave steepening. The nonlinear mode will be used where the simulation of wave steepening is judged to be important, albeit at the cost of a small loss of mass. A multilinear method, representing a compromise between linear and nonlinear methods, has been developed recently by Perumal (1992). The MuskingumCunge is a viable alternative to the classical Muskingum method, particularly for the cases where hydrologic data (i.e., streamflow data) are not available, but where hydraulic data (crosssectional data and channel slopes) can be readily ascertained. In many instances, the MuskingumCunge method is also an alternative to the more complex dynamic wave models, which lack robustness and have significant data requirements. The nonlinear features of the variableparameter MuskingumGunge method makes it the method of choice in hydrologic flood routing. This paper revisits the variableparameter MuskingumCunge method of Ponce and Yevjevich (1978). The small but perceptible loss of mass is confirmed throughout a wide range of unit peak inflows, from 200 to 1000 cfs ft^{1} (18.5892.9 m^{2}s^{1}). Furthermore, a slight improvement in mass conservation is realized by an alternative way of calculating the variable parameters.
2. BACKGROUND
The MuskingumCunge method is a variant of the Muskingum method (Chow, 1959) developed by Cunge (1969) and documented in the Flood Studies Report (Natural Environment Research Council, 1975). Ponce and Yevjevich (1978) expressed the routing parameters of the MuskingumCunge method in terms of the Courant and cell Reynolds numbers, two physically and numerically meaningful parameters. In addition, they developed a nonlinear mode of calculating the parameters, thereby enhancing the method's applicability to realworld routing problems. A threepoint method and an iterative fourpoint method were suggested as a way to vary the parameters as a function of local flow values. The method has been adopted by the most recent version (Version 4.0) of the US Corps of Engineers' HEC1 model (1990). The routing equation of the MuskingumCunge method defined in the typical fourpoint grid configuration is:
in which j is a spatial index, n is a temporal index, and:
C_{0} = (1 + C + D) / (1 + C + D)
C_{1} = (1 + C  D) / (1 + C + D)
C_{2} = (1  C + D) / (1 + C + D)
are the routing coefficients, with C the Courant number, defined as follows:
C = c (Δt / Δx)
in which c is the wave cerelity, Δt is the time interval, Δx is the space interval, and D the cell Reynolds number, defined as follows:
D = q /(S_{o} c Δx)
in which q is the unitwidth discharge and S_{o} is the bottom slope. The wave celerity is defined as follows:
c = β (Q /A) = β (q / d )
in which β is the exponent of the rating, A is the flow area, and d is the flow depth. The MuskingumCunge method matches the numerical diffusion of the discrete model with the physical diffusion of the analytical model. The advantage is that the routing results are independent of the grid specification, provided numerical dispersion is minimized. The numerical dispersion can be minimized by keeping the Courant number equal to or slightly greater than one (Cunge, 1969; Ponce and Theurer, 1982; Ferrick et. al., 1983).
3. NUMERICAL EXPERIMENTS
In this paper, Thomas' (1934) classical problem was chosen to test the MuskingumCunge method under a wide range of flow conditions. The Thomas problem was chosen here to allow comparison with previous results (Ponce and Yevjevich, 1978). The original problem considered routing a sinusoidal flood wave with a 96h period through a prismatic channel 500 miles long. A unitwidth channel of baseflow q_{b} = 50 cfs ft^{1} (4.64 m^{2} s^{1}) and peak inflow q_{pi} = 200 cfs ft^{1} (18.58 m^{2} s^{1}) was specified. The bottom slope was set at 1 ft mile^{1}, and the dischargedepth rating was the following:
q = 0.688 d ^{5/3}
This amounts to a Manning's n = 0.0297. It can be shown that the 96h sinusoidal flood wave satisfies the diffusion wave applicability criterion (Ponce et. al., 1978). The following methods are considered here:
Two levels of spatial and temporal resolution were chosen:
Three peak inflow to baseflow ratios q_{pi} /q_{b} were used:
These peak
inflow to baseflow ratios encompass a wide range of practical values. With
q_{pi} =
4. RESULTS
The mass conservation properties were assessed by integrating inflow and outflow
hydrographs for the 500mile reach. Table 1 shows peak outflow q_{po}, ratio of peak
outflow to peak inflow
The following conclusions are drawn from Table 1:
It should be noted that the mass conservation percentages shown in Column 7 of Table 1 are cumulative values for the 500mile reach of the Thomas problem. A typical flood routing application would normally not consider such long reach without intervening lateral inflows which tend to mask the accuracy of the computation. Therefore, the percentages shown should be interpreted as an extreme or worstcase scenario.
Figure 1 shows a set of flood hydrographs for resolution level II and peak inflow to
baseflow
5. CONCLUSIONS The modified variableparameter methods MVPMC3 and MVPMC4 developed in this paper result in a definite improvement in mass conservation, as compared with the conventional methods VPMC3 and VPMC4 (Ponce and Yevjevich, 1978). The improvement may be more marked when using a wide range of realistic discharges. Otherwise, the small loss of mass does not appear to impose a significant drawback. It is the price to be paid to model the nonlinear features of a flood wave within the framework of the MuskingumCunge method. REFERENCES Chow, V. T., 1959. OpenChannel Hydraulics. McGrawHill, New York. Cunge, J. A., 1969. On the subject of a flood propagation computation method (Muskingum method). J. Hydraul. Res., 7(2) 205230. Dooge, J. C. I., 1973. Linear theory of hydrologic systems. U.S. Dep. Agric. Tech. Bull., 1468, 327 pp. Ferrick, M. G., Bilmes, J. and Long, S. E., 1983. Modeling rapidly varied flow in tailwaters. Water Resour. Res., 20(2): 271289. Natural Environment Research Council, 1975. Flood Studies Report, Vol. III. Natural Environment Research Council, London. Perumal, M., 1992. Multilinear Muskingum flood routing method. J. Hydrol., 133: 259272. Ponce, V. M. and Theurer, F. D., 1982. Accuracy criteria in diffusion routing. J. Hydraul. Div., ASCE., 108 (HY6): 747757. Ponce, V. M. and Yevjevich, V., 1978. MuskingumCunge method with variable parameters. J. Hydraul. Div. ASCE, 104(HY12): 16631667. Ponce, V. M., Li, RM. and Simons, D. B., 1978. Applicability of knematic and diffusion models. J. Hydraul. Div. ASCE, 104(HY3): 353360. Thomas, H. A., 1934. The hydraulics of flood movement in rivers. Eng. Bull., Carnegie Inst. of Technology, Pittsburgh, PA. US Army Corps of Engineers, 1990. HEC1, flood hydrograph package: User's manual, version 4, 1990. US Army Corps of Engineers, Hydrologic Engineering Center, Davis, CA.

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