عنوان مقاله [English]
Analysis of geometric order of drainage networks using Tokunaga model and capacity dimension
(Case study: Bashar river basin)
The Tokunaga orders for drainage networks have been discussed extensively in the geomorphology and nicely reviewed. The branching structure of drainage networks has been actively studied since the 1980s, to address a broad range of geomorphological, hydrological, geological and environmental problems. The hierarchical organization and self-similarity in drainage networks have been topics of extensive research in geomorphology and hydrology starting with the pioneering work of Tokunaga in 1945. The self-similarity and fractal scaling in drainage networks have been topics of extensive research in hydrology and geomorphology. (Durighetto et al, 2020; Yan et al, 2018). Here we capitalize on a recently developed theory of random self-similar trees to elucidate the origin of Tokunaga model, Horton-Strahler ordering and basin fractal dimensions. Indeed, different basin properties may affect how drainage networks are morphologically structured such as climate, geology, soil composition, sediment, topography and vegetation. It allows one to quantitatively describe the connections between the river streams properties and various observed processes that operate on the river networks, as well as to perform comprehensive numerical simulations aimed at hypothesis testing and improving the understanding of the river streams dynamics. For example, the observed scaling relationships between geomorphic (e.g., drainage networks) and hydrologic (e.g., annual peak flow) variables have been frequently studied through ensemble simulations of river streams (Joshi & Kotlia, 2018;Yan et al, 2018; Liao et al, 2019). In this paper, the Tokunaga model and the fractal correlation dimension have been used to investigate the drainage network of Bashar basin.
Bashar River is one of the sub-basins of Karoon watershed located in the southwest of the country. It is located in a mountainous area and it is snowing and raining. This river originates from the northwestern heights of Ardakan city in Fars province and flows through deep valleys in a northwesterly direction. The average rainfall of the basin is 858 mm. The average river discharge is 53.4 cubic meters per second and the maximum and minimum discharge are 106.5 and 18.55 cubic meters per second, respectively. Tokunaga model is built on the assumption of self-similarity in the side tributary structure, and an additional assumption of Tokunaga self-similarity, which is supported by data from drainage networks. The Tokunaga model extends the Horton-Strahler ordering by cataloging side-branching, that is the merging of streams of different orders; it is illustrated in (Fig. 1). At each level the primary branch, say 22, has binary branching, two 11 branches, and one 12 side branch. This simple branching structure can be extended to all orders. This is the basic concept of the Tokunaga side branching. Tokunaga ordering with side branching is illustrated in (Fig. 1). (Newman et al, 1997; Zanardo et al, 2013).
Fig. 1. Example of (a) Horton-Strahler ordering, and of (b) Tokunaga model.
Results and Discussion
Results show that analytical proof required no restrictions on the parameters a and c, other than that they be positive real numbers. In practice, however, parameters a and c should be close to 1 and 2, respectively, to predict a value of the bifurcation ratio for river streams between 3 and 5 as data suggests (fig. 2). Therefore, it is reasonable to expect some physical restrictions on parameters a and c, which is a topic for future research. That drainage networks on the average behave in much the same way as the Tokunaga model suggests that the parameters a and c contain information that is fundamental to the understanding of river streams evolution from geomorphological processes of erosion and runoff generation. Prediction of Tokunaga parameters a and c from physical processes is a very important topic for future research.
Fig. 1. Dependence of T (K) with constant values of a = 0.08 and c = 3.5
Tokunaga model is gaining increasing recognition in geomorphology and hydrology, because data supports several predictions of the model. We conclude by suggesting that the understanding of the Tokunaga side branching and self-similarity in river streams gained here can be extended to other hydrological and geomorphological processes. Important examples include fractal scaling and drainage networks. River streams satisfy fractal scaling laws in a variety of ways. It is concluded that the models developed in hydrology have direct applicability to the fundamental problems in geomorphology. For river streams the bifurcation ratio is fractal both in terms of the Tokunaga side branching and the Horton–Strahler primary branching.
Keywords: Capacity dimension, geometric order, Tokunaga model, drainage network, Bashar Basin.