Scaling mean annual peak flow scaling with upstream basin area

Understanding how annual peak flow, $Q_p$, relates to upstream basin area, $A$, and their scaling have been one of the challenges in surface hydrology. Although a power-law scaling relationship (i.e., $Q_p \propto A^\alpha$) has been widely applied in the literature, it is purely empirical, and due to its empiricism the interpretation of its exponent, a, and its variations from one basin to another is not clear. In the literature, different values of a have been reported for various datasets and drainage basins of different areas. Invoking concepts of percolation theory as well as self-affinity, we derived universal and non-universal scaling laws to theoretically link $Q_p$ to $A$. In the universal scaling, we related the exponent $\alpha$ to the fractal dimensionality of percolation, $D_x$. In the non-universal scaling, in addition to $D_x$, the exponent a was related to the Hurst exponent, $H$, characterizing the boundaries of the drainage basin. The $D_x$ depends on the dimensionality of the drainage system (e.g., two or three dimensions) and percolation class (e.g., random or invasion percolation). We demonstrated that the theoretical universal and non-universal bounds were in well agreement with experimental ranges of a reported in the literature. More importantly, our theoretical framework revealed that greater a values are theoretically expected when basins are more quasi two-dimensional, while smaller values when basins are mainly quasi three-dimensional. This is well consistent with the experimental data. We attributed it to the fact that small basins most probably display quasi-two-dimensional topography, while large basins quasi-three-dimensional one.