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An analytical theoryis developed that obtains Horton laws for six hydraulic–geometric (H–G)variables (stream discharge Q, width W, depth D, velocity U, slopeS, and friction n') in self-similar Tokunaga networks in the limit of alarge network order. The theory uses several disjoint theoretical conceptslike Horton laws of stream numbers and areas as asymptotic relations inTokunaga networks, dimensional analysis, the Buckingham Pi theorem,asymptotic self-similarity of the first kind, or SS-1, and asymptoticself-similarity of the second kind, or SS-2. A self-contained review of theseconcepts, with examples, is given as "methods". The H–G data sets inchannel networks from three published studies and one unpublished study aresummarized to test theoretical predictions. The theory builds on sixindependent dimensionless river-basin numbers. A mass conservationequation in terms of Horton bifurcation and discharge ratios in Tokunaganetworks is derived. Assuming that the H–G variables are homogeneous andself-similar functions of stream discharge, it is shown that the functionsare of a power law form. SS-1 is applied to predict the Horton laws forwidth, depth and velocity as asymptotic relationships. Exponents of width andthe Reynolds number are predicted and tested against three field data sets.One basin shows deviations from theoretical predictions. Tentatively assumingthat SS-1 is valid for slope, depth and velocity, corresponding Horton lawsand the H–G exponents are derived. Our predictions of the exponents are thesame as those previously predicted for the optimal channel network (OCN)model. In direct contrast to our work, the OCN model does not consider Hortonlaws for the H–G variables, and uses optimality assumptions. The predictedexponents deviate substantially from the values obtained from three fieldstudies, which suggests that H–G in networks does not obey SS-1. It failsbecause slope, a dimensionless river-basin number, goes to 0 as network orderincreases, but, it cannot be eliminated from the asymptotic limit. Therefore,a generalization of SS-1, based on SS-2, is considered. It introduces twoanomalous scaling exponents as free parameters, which enables us to show theexistence of Horton laws for channel depth, velocity, slope and Manningfriction. These two exponents are not predicted here. Instead, we used theobserved exponents of depth and slope to predict the Manning frictionexponent and to test it against field exponents from three studies. The samebasin mentioned above shows some deviation from the theoretical prediction. Aphysical reason for this deviation is given, which identifies an importanttopic for research. Finally, we briefly sketch how the two anomalous scalingexponents could be estimated from the transport of suspended sediment loadand the bed load. Statistical variability in the Horton laws for the H–Gvariables is also discussed. Both are important open problems for futureresearch

Horton laws for hydraulic–geometric variables and their scaling exponents in self-similar Tokunaga river networks