Approximate convection-diffusion equations

Abstract

This paper describes the development of simplified momentum equations, in stage as well as in discharge formulations, governing the transition between the diffusion and the kinematic waves (including the latter). It also describes the application of these equations to arrive at the approximate convection-diffusion equations. The appropriateness of these approximate convection-diffusion equations to model flood waves in the transition range is established, and the characteristics of these equations are studied from the point of view of description of the loop-rating curve. The development of these simplified equations provides a theoretical justification for their use in the well-known 'Jones formula' expressed as Q/Q0 = {1 + [1/(S0c)]partial differentialy/partial differentialt}( 1/2 ) for converting the stage to discharge of a diffusive flood wave-an approach that has hitherto been considered to be logically incorrect.This paper describes the development of simplified momentum equations, in stage as well as in discharge formulations, governing the transition between the diffusion and the kinematic waves (including the latter). It also describes the application of these equations to arrive at the approximate convection-diffusion equations. The appropriateness of these approximate convection-diffusion equations to model flood waves in the transition range is established, and the characteristics of these equations are studied from the point of view of description of the loop-rating curve. The development of these simplified equations provides a theoretical justification for their use in the well-known 'Jones formula' expressed as Q/Q0 = {1 + [1/(S0c)]∂y/∂t}1/2 for converting the stage to discharge of a diffusive flood wave - an approach that has hitherto been considered to be logically incorrect.