Scale Invariance and Self-Similarity in Hydrologic Processes in Space and Time

There is a strong analogy between fractal geometries and scale invariant processes. Fractal geometries are self-similar at different scales. Similar to fractal geometries, solutions of scale invariant processes at different space-time scales are self-similar. This unique property of scale-invariant processes can be employed to find the solution of the processes at a much larger or smaller space-time scale based on the solution calculated on the original scale. Here, we investigate scale invariance properties of hydrologic processes as initial-boundary value problems in one-parameter Lie group of point transformations framework. Scaling (stretching) transformation has unique importance among other Lie group of point transformations, as it leads to the scale invariance or scale dependence of a process. Scale invariance of a process allows using the same mathematical model for the process at different scales and facilitates finding the solution at any scale using the solution at the original scale. In this study, the process parameters and source/sink terms are regarded as state variables of some (secondary) processes that underlie or couple with the original process. Then under the scaling transformations, the invariance conditions for the resulting system of processes at time-space scales that are different from the original time-space scales are investigated. The conditions to be satisfied by the form of a governing equation and its parameters, as well as the initial and boundary conditions of the process, are established in order for the process to be scale invariant. Also, the self-similarity of the solution of an invariant process is demonstrated by various numerical example problems.