Moments and cumulants of linearized St. Venant equation Renata J. Romanowiez Institute of Geophysics, Polish Academy of Sciences, Warsaw, Pasteura 3, Poland James C. I. Dooge Department of Civil Engineering, University College Dublin, Earlsfort Terrace, Dublin 2, Ireland and Zbigniew W. Kundzewicz Institute of Geophysics, Polish Academy of Sciences, Warsaw, Pasteura 3, Poland A general closed form expression for the Rth cumulant of the unsteady flow due to an upstream impulse input in a semi-infinite channel is derived by (a) reducing the recurrence relationship between successive cumulants to a recurrence relationship between the set of parameters 7(R, i) characterizing the ratio of successive terms in the series for the Rth cumulant and (b) deriving the closed form equivalent to this recurrence relationship in terms of either nested sums or factorials. Key Words: cumulant, St. Venant equation, linear flood routing 1. INTRODUCTION One-dimensional analysis of unsteady flow in open channels is based on the continuity equation and the momentum equation in the average direction of flow as formulated by St. Venant ~. Though the continuity equation is linear in discharge Q(x, t) and area of flow A(x, t), the momentum equation is highly nonlinear and hence no general analytical solution is available. Numerical solution is possible but must be carried out separately for each set of channel characteristics and each set of boundary conditions. Strictly speaking, each such separate computation requires an individual sensitivity analysis if the claim is made that the level of error of computational approximation is appropriate to the accuracy of the field data. In all branches of mathematical physics, the strategy is followed of proposing simplified models and comparing their predictions with both reliable field data and with carefully controlled numerical predictions for a wide range of representative cases. The aim of such a strategy is twofold: to produce convenient reliable predictive models and to assist in the understanding of the basic physical phenomena. The most common starting point in such a strategy is the linearization of a set of nonlinear equations said to simulate adequately the physical phenomena under study. It is not surprising therefore that the linearized form of the St. Venant equation has been used both in the comparative study of predictive methods of flood routing and in the study of importance of phenomena displayed by field data. The first direct application of linearization to the St. Venant equations was made 50 years ago by Deymie z and Masse 3. The same result was obtained independently by Dooge and Harley4's who made use as a diagnostic tool of moments and cumulants used so widely as such in statistics and linear system theory. The linearized St. Venant equation has been applied to such © Computational Mechanics Publications 1988 92 Adv. Water Resources, 1988, Volume 11, June problems as flood routing in rivers 6, systematic comparison of hydrologic routing methods 5'7, multi- linear model for flood routing 8, simulation of the holding time in the geomorphic unit hydrograph 9, the effect of the downstream boundary conditions on flood routing 1°-13 and the harmonic analysis of flood movements in rivers 14. It was shown by Dooge and Harley4 and by Strupczewski and Kundzewicz7 that the known expressions for the first four moments and cumulants could be used to deduce certain properties of the linearized solutions and of the errors of linearization involved. The sensitivity of the Rth cumulant to the reference velocity (Uo) and hence the basic error of linearization can be shown to be given by kn ~/,/~)(R) where :~(R) is an increasing function of R given by R(2 - m)- 1 • (R) = m-1 where m is the ratio of the kinematic wave speed to average velocity of flow and depends only on the shape of the channel section and the friction law used 15"16. The higher cumulants of the channel response and hence of the output are only excited by the equivalent order polynomials in the input function4. It can be reasonably hypothesized that for a bounded input the relative contributions of the higher order polynomials will be relatively small and that for a stable condition of unsteady flow (i.e., no roll waves or hydraulic bores) the growth in the sensitivity of the higher cumulants will not be sufficient to produce growing higher order terms in the output. To prove such a hypothesis analytically or to improve its strength by carefully controlled numerical experiments requires a general expression for the Rth