Geometry of River Networks I: Scaling, Fluctuations, and Deviations

This article is the first in a series of three papers investigating the detailed geometry of river networks. Large-scale river networks mark an important class of two-dimensional branching networks, being not only of intrinsic interest but also a pervasive natural phenomenon. In the description of river network structure, scaling laws are uniformly observed. Reported values of scaling exponents vary suggesting that no unique set of scaling exponents exists. To improve this current understanding of scaling in river networks and to provide a fuller description of branching network structure, we report here a theoretical and empirical study of fluctuations about and deviations from scaling. We examine data for continent-scale river networks such as the Mississippi and the Amazon and draw inspiration from a simple model of directed, random networks. We center our investigations on the scaling of the length of sub-basin's dominant stream with its area, a characterization of basin shape known as Hack's law. We generalize this relationship to a joint probability density and show that fluctuations about scaling are substantial. We find strong deviations from scaling at small scales which can be explained by the existence of linear network structure. At intermediate scales, we find slow drifts in exponent values indicating that scaling is only approximately obeyed and that universality remains indeterminate. At large scales, we observe a breakdown in scaling due to decreasing sample space and correlations with overall basin shape. The extent of approximate scaling is significantly restricted by these deviations and will not be improved by increases in network resolution.Comment: 16 pages, 13 figures, Revtex4, submitted to PR

Unified View of Scaling Laws for River Networks

Scaling laws that describe the structure of river networks are shown to follow from three simple assumptions. These assumptions are: (1) river networks are structurally self-similar, (2) single channels are self-affine, and (3) overland flow into channels occurs over a characteristic distance (drainage density is uniform). We obtain a complete set of scaling relations connecting the exponents of these scaling laws and find that only two of these exponents are independent. We further demonstrate that the two predominant descriptions of network structure (Tokunaga's law and Horton's laws) are equivalent in the case of landscapes with uniform drainage density. The results are tested with data from both real landscapes and a special class of random networks.Comment: 14 pages, 9 figures, 4 tables (converted to Revtex4, PRE ref added

Regio- and Enantioselective Formal Hydroamination of Enamines for the Synthesis of 1,2-Diamines

The asymmetric formal hydroamination of enamines using a CuH catalyst is reported. The method provides a straightforward and efficient approach to the synthesis of chiral 1,2-dialkyl amines in good yields with high levels of enantioselectivities for a broad range of substrates, and should have significant value for the preparation of molecules bearing a 1,2-diamine motif

Determination of the relative configuration of vic-amino alcohols

A large set of syn- and anti-1,2-amino alcs. has been synthesized. Upon comparison of the CHO and CHN 1H NMR shifts a general trend has been obsd., making it possible to det. the relative configuration of 1,2-amino alcs. without derivatization or shift reagents

Synthesis of Imidates : TFA-Mediated Regioselective Amide Alkylation Using Meerwein's Reagent

Regioselective O-alkylation of an amide to form the corresponding imidate is a common synthetic problem, often resulting in varying amounts of N-alkylation. Screening existing methods for converting amides to imidates gave inconsistent or irreproducible results, sometimes affording N-alkylamide as the major product. A simple and reliable protocol for amide O-alkylation with complete regioselectivity has been designed, and its scope and efficiency demonstrated on a number of substrates

Enantioselective synthesis of anti-beta-amido-alpha-hydroxy esters via asymmetric transfer hydrogenation coupled with dynamic kinetic resolution

The asymmetric transfer hydrogenation of beta-amido-alpha-keto esters providing the corresponding anti-beta-amido-alpha-hydroxy esters via dynamic kinetic resolution is reported. The use of a commercially available, or simply prepared, chiral ruthenium catalyst results in good yields as well as high diastereoselectivities and enantioselectivities. (C) 2013 Elsevier Ltd. All rights reserved