Symmetry in Nonlinear Hydrologic Dynamics under Uncertainty: Modeling Approach

Symmetry methods can be used to transform almost any kind of linear or nonlinear partial differential equation (PDE) that represents a hydrologic process in any dimension to an equivalent ordinary differential equation (ODE). Meanwhile, Kavvas recently shown in 2003 that the conservation equations of hydrologic processes under uncertainty, expressed as linear or nonlinear stochastic ODEs or PDEs, have a one-to-one correspondence to a mixed Eulerian-Lagrangian nonlocal form of the Fokker-Planck equation (FPE) when the underlying process has finite correlation lengths. Under such correspondence, it is possible to obtain a solution for the ensemble behavior of a particular hydrologic process in terms of the solution of its corresponding FPE for the probability distribution function (PDF) of its state variables under appropriate initial and boundary conditions. A major issue with the resulting FPE in the case of conservation equations in PDE form is that spatial gradients of the process state variables appear in the resulting FPE that prevent its solution. Therefore, a formal algorithm is needed to reduce the PDE of a hydrologic process into an ODE in order to eliminate the original spatial gradients of the process state variables in the corresponding FPE of the process. This is accomplished by the symmetry methods. After such transformation, the resulting FPE, which is a linear deterministic PDE, can be solved to obtain the evolutionary PDF, ensemble average, ensemble variance, and any other statistical function of the hydrologic process being investigated.