Hydraulic geometry and minimum rate of energy dissipation

Abstract

The theory of minimum rate of energy dissipation states that a system is in an equilibrium condition when its rate of energy dissipation is at its minimum value. This minimum value depends on the constraints applied to the system. When a system is not at equilibrium, it will adjust in such a manner that the rate of energy dissipation can be reduced until it reaches the minimum and regains equilibrium. A river system constantly adjusts itself in response to varying constraints in such a manner that the rate of energy dissipation approaches a minimum value and thus moves toward an equilibrium. It is shown that the values of the exponents of the hydraulic geometry relationships proposed by Leopold and Maddock for rivers can be obtained from the application of the theory of minimum rate of energy dissipation in conjunction with the Manning‐Strickler equation and the dimensionless unit stream power equation for sediment transport proposed by Yang. Theoretical analysis is limited to channels which are approximately rectangular in shape. The width and depth exponents thus derived agree very well with those measured in laboratory experiments by Barr et al. Although the theoretical width and depth exponents are within the range of variations of measured data from river gaging stations, the at‐station width adjustment of natural rivers may also depend on constraints other than water discharge and sediment load.