# Applied Math Seminar: Horton Ratios Link Self-similarity with Maximum Entropy of Eco-Geomorphological Properties in Stream Networks

Event Type:

Seminar

Speaker:

Bruce Milne

Event Date:

Monday, April 3, 2017 - 3:30pm

Location:

SMLC 356

### Event Description:

Author: Milne(1), Bruce T., and Vijay K. Gupta(2)

Affiliation: (1)University of New Mexico, Albuquerque, NM; (2)University of Colorado, Boulder

Stream networks are highly branched structures wherein water and energy move between land and atmosphere, modulated by evapotranspiration and its interaction with the gravitational dissipation of potential energy. Stream networks are self-similar, as reflected in the ubiquitous Horton laws for the Horton ratios of stream numbers of various Strahler orders, associated basin areas, and stream lengths. Recently, Horton ratios were demonstrated for ecological phenomena including riparian tree species richness and Shannon diversity along 13,062 streams in Whitewater basin, Kansas. We demonstrate that Horton ratios are equivalent to Lagrange multipliers used in the extremum function leading to Shannon entropy being maximal, subject to constraints. Some properties, such as basin area, reach total entropy expected for an unconstrained system, while the entropies of others are constrained. We suggest that properties of stream networks vary in accordance with constraints imposed by stream runoff and sediment, land use, and inter-annual variation in water balance that challenges trees to find an optimal root-to-shoot ratio to match expected annual precipitation. Synthesis of Horton laws as scaling relations and maximum entropy provides a framework for understanding stream networks as interactions of constituent parts that reconcile fluxes of water and energy, thereby forming the whole. The entropy-Horton framework may inform assessments of system resilience to perturbations in water supply, potential evapotranspiration, invasive species, and land use changes.

### Event Contact

**Contact Name: **Deborah Sulsky