Dimensionally Consistent Nonlinear Muskingum Equation

Abstract

Although the Muskingum equation was proposed nearly 75 years ago, it is still a subject of active research. Despite of its simple form, the real properties of this equation have not been comprehensively explained. This paper proposes a new interpretation of the linear McCarthy’s relation. This relation can be interpreted only together with the storage equation, whereas the Muskingum equation can be derived directly from the system of equations describing the kinematic wave model. Because the Muskingum equation is a semidiscrete form of the kinematic wave equation, all typical effects which may occur in its numerical solution, such as pure translation, attenuation, and oscillations, can be comprehensively explained using the modified equation approach. Taking into account the kinematic roots of the linear Muskingum equation, its new nonlinear version is derived without using any additional McCarthy-type formula. It is shown that the proposed equation is the only correct conservative form of the nonlinear Muskingum equation which simultaneously ensures its dimensional consistency.