Nonlinear Hydrologic Processes: Conservation Equations for Determining Their Means and Probability Distributions

The point-location-scale conservation equations of hydrologic processes, when viewed at the scale of computational grid areas, become stochastic partial differential equations (PDEs). For the upscaling of the point-location-scale conservation equations to the scale of computational grid areas, a common approach is to develop the ensemble averages of these equations. Accordingly, in this study general ensemble average conservation equations for determining the probabilistic and mean behavior of nonlinear and linear hydrologic processes are developed to exact second order. From the derived equations it is seen that the evolution equation for the probabilistic behavior of a generally nonlinear hydrologic system becomes a Fokker–Planck equation (FPE). As such, the determination of the probabilistic behavior of a hydrologic system of processes reduces to the solution of a linear, deterministic PDE, the FPE, under appropriate initial and boundary conditions. The solution of the FPE yields the probability density function of the hydrologic system which can then be used to obtain the means of the state variables of the system by the expectation operation. One can also determine the mean behavior of nonlinear stochastic hydrologic processes by means of master key ensemble average conservation equations developed in this study. Upon examination of these generic deterministic equations, one may note that they are implicit integro-differential nonlinear equations in the mixed Eulerian–Lagrangian form. Meanwhile, the master key equations which were developed also for linear hydrologic processes, are explicit, linear PDEs.