Fig. 1  Flood on the Cuiaba river, Mato Grosso, Brazil, on January 10, 1995. NEW PERSPECTIVE ON THE CONVECTION-DIFFUSION-DISPERSION EQUATION Victor M. Ponce Online version 2021 [Original version 1994] Abstract. The coefficients of the dimensionless partial differential equation of convection-diffusion-dispersion of flood waves are shown to be functions of the Froude and Vedernikov numbers only. The Froude number is the ratio of mean velocity to relative dynamic wave celerity. The Vedernikov number is the ratio of relative kinematic wave celerity to relative dynamic wave celerity. The third-order convection-diffusion-dispersion equation may be useful in the analysis of flood propagation problems where both diffusion and dispersion are significant. 1. INTRODUCTION The convection-diffusion model of flood waves [Hayami, 1951; Dooge, 1973] is improved with the addition of the third-order dispersion term, in effect constituting a convection-diffusion-dispersion model [Ferrick et al., 1984]. With the appropriate dimensional technique, the coefficients of the resulting third-order partial differential equation can be shown to be a function only of the Froude and Vedernikov numbers. This underscores the importance of these two dimensionless numbers in describing the dynamics of flood wave propagation. 2. CONVECTION-DIFFUSION-DISPERSION EQUATION The convection-diffusion model of flood waves is originally due to Hayami [1951]. Dooge [1973] improved the Hayami model by including inertia in the formulation of the diffusion coefficient, resulting in a convection-diffusion model with inertia. Dooge et al. [1982] generalized the convection-diffusion model with inertia for any friction type and cross-sectional shape. Ponce [1991] expressed the diffusion coefficient in terms of the Vedernikov number. Ferrick et al. [1984] derived the most complete third-order linear equation to date, which includes convection, diffusion, and dispersion processes, to wit: Qt + c Qx = ν Qxx + η Qxxx (1) in which Q = discharge, c = convective celerity, ν = diffusion coefficient, and η = dispersion coefficient. The convective celerity, or flood wave speed, is defined as follows [Seddon, 1900; Chow, 1959]:         dQ c = _____         dA (2) in which A = flow area. The diffusion coefficient, or hydraulic diffusivity, for Chezy friction in hydraulically wide channels is [Dooge, 1973]:           qo              F 2 ν = ______ ( 1 - _____ )         2 So              4 (3) in which qo = unit-width discharge, So = bottom slope, and F = Froude number, defined as follows [Chow, 1959]:             uo F = __________        (g Do)1/2 (4) in which uo = mean velocity, Do = hydraulic depth, and g = gravitational acceleration. In most cases of practical interest, the hydraulic depth may be approximated by the flow depth yo. The dispersion coefficient, or hydraulic dispersivity, is the following [Ferrick et al., 1984]:                   yo η = F 2 ( ______ ) v                  2So (5) Ponce [1991] has expressed the convective celerity as a function of the Froude and Vedernikov numbers, as shown by Eq. 6. The Vedernikov number is the ratio of relative kinematic and dynamic wave celerities [Craya, 1952].                  V c = ( 1 + ____ ) uo                  F (6) Following Dooge et al. [1982], Ponce [1991] expressed the diffusion coefficient in terms of the Vedernikov number, in effect generalizing it for all friction types (turbulent Manning or Chezy, and laminar) and cross-sectional shapes, including hydraulically wide, triangular, and inherently stable [Ponce and Porras, 1995]:          qo ν = ______ ( 1 - V 2 )         2 So (7) in which [Dooge et al. 1982; Ponce 1991]: V = ( β - 1 ) F (8) with β = exponent of the discharge-flow area rating Q = α Aβ. Given qo = uoyo, and defining a reference channel length Lo = yo /So [Lighthill and Whitham, 1955], the diffusion coefficient is:           Lo ν = ( ____ ) uo ( 1 - V 2 )            2 (9) Furthermore, with Eq. 9, the dispersion coefficient is:            Lo η = ( ____ ) 2 uo ( 1 - V 2 ) F 2             2 (10) 3. DIMENSIONLESS CONVECTION-DIFFUSION-DISPERSION EQUATION To nondimensionalize Eq. 1, we choose appropriate dimensionless variables such that x' = x /Lo, and t' = t (uo /Lo). Then, Eq. 1 converts to: Qt'  + c' Qx'  = ν' Qx'x'  + η' Qx'x'x' (11) in which c' = dimensionless celerity:                V c' = 1 + _____                F (12) ν' = dimensionless diffusivity:          1 ν' = ___ ( 1 - V 2 )          2 (13) and η' = dimensionless dispersivity:          1 η' = ___ ( 1 - V 2 ) F 2          4 (14) Therefore, the three coefficients of the dimensionless convection-diffusion-dispersion equation (Eqs. 12 to 14) are shown to be only functions of the Froude and Vedernikov numbers. The online calculator ONLINEDISPERSIVITY.PHP calculates the following wave properties: (1) celerity, (2) diffusivity, and (3) dispersivity. Input is based in the following variables: (a) mean velocity uo, (b) hydraulic depth Do, which may be approximated as flow depth yo, (c) channel slope So, and (d) exponent of the rating β. 4. SUMMARY The coefficients of the dimensionless partial differential equation of convection-diffusion-dispersion of flood waves are shown to be functions only of the Froude and Vedernikov numbers. The Froude number is the ratio of mean velocity to relative dynamic wave celerity. The Vedernikov number is the ratio of relative kinematic wave celerity to relative dynamic wave celerity. The third-order convection-diffusion-dispersion equation may be useful in the analysis of flood propagation problems where both diffusion and dispersion are significant. REFERENCES Chow, V. T. 1959. Open-channel Hydraulics, McGraw-Hill. New York. Craya, A. 1952. The criterion for the possibility of roll wave formation. In Gravity Waves, Circ. 521, pp. 141-151, National Institute of Standards and Technology, Gaithersburg, MD. Dooge, J. C. I. 1973. Linear theory of hydrologic systems: Chapter 9. Technical Bulletin No. 1468, 327 pp., U.S. Department of Agriculture, Washington. D.C. Dooge, J. C. I., W. B. Strupczewski, and J. J. Napiorkowski. 1982. Hydrodynamic derivation of storage parameters of the Muskingum model. Journal of Hydrology, 54, 371-387. Ferrick, M. G., J. Bilmes, and S. E. Long. 1984. Modeling rapidly varied flow in tailwaters. Water Resources Research, 20(2), 271-289. Hayami, S., On the propagation of flood waves. 1951. Bulletin Disaster Prevention Reserach Institute, Kyoto University. Japan, 1, 1-16. Lightbill, M. J., and G. B. Whitman. 1955. On kinematic waves. I. Flood movement in long rivers. Proceedings of the Royal Society of London, A, 229, 281-316. Ponce, V. M. 1991. New perspective on the Vedernikov number. Water Resources Research, 27(7), 1777-1779. Ponce, V. M. and P. J. Porras. 1995. Effect of cross-sectional shape on free-surface instability. ASCE Journal of Hydraulic Engineering, Vol. 121, No. 4, April, 376-380. Seddon, J. A. 1900. River hydraulics. Transactions, American Society of Civil Enginners, 43, 179-243.