Abstract
Abstract. By rigorously accounting for dimensional homogeneity in physical laws, the Π theorem and the related self-similarity hypotheses allow us to achieve a dimensionless reformulation of scientific hypotheses in a lower-dimensional context. This paper presents applications of these concepts to the partitioning of water and soil on terrestrial landscapes. For such processes, their complexity and lack of first principle formulation make dimensional analysis an excellent tool to formulate theories that are amenable to empirical testing and analytical developments. The resulting scaling laws help reveal the dominant environmental controls for these partitionings. In particular, we discuss how the dryness index and the storage index affect the long-term rainfall partitioning, the key nonlinear control of the dryness index in global datasets of weathering rates, and the existence of new macroscopic relations among average variables in landscape evolution statistics. The scaling laws for the partitioning of sediments, the elevation profile, and the spectral scaling of self-similar topographies also unveil tantalizing analogies with turbulent flows.
Highlights
Galileo is credited as the first scientist to have used dimensional analysis and scaling
We focus on evapotranspiration rate (ET), as many have done before, but here we adopt the special lens offered by the theorem to explore the implications of different hypotheses in the physical laws used as starting points
It is time to draw to a close and ask ourselves whether by using dimensional analysis we have learned anything useful regarding these problems
Summary
Galileo is credited as the first scientist to have used dimensional analysis and scaling. In his 1638 “Dialogues Concerning Two New Sciences” (Galilei, 1914), he deduced that geometrically similar objects are not strong under their own weight: “A small dog could probably carry on his back two or three dogs of his own size; but I believe that a horse could not carry even one of his own size” Since this discovery of scaling laws for complex biological materials, dimensional analysis has continued to fascinate many scientists, from Fourier and Maxwell to Reynolds, Rayleigh, Kolmogorov, and Taylor, and contributed to numerous new results in several fields (e.g., Barenblatt, 1996; Szirtes, 2007; Bolster et al, 2011; Katul et al, 2019).
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