The use of power law forms to describe hydraulic geometry is a classic subject with a history of over 70 years. Two distinct forms of power laws have been proposed: at-a-station hydraulic geometry (AHG) and downstream hydraulic geometry (DHG). Although the utility of these semiempirical expressions is widely recognized, they remain poorly understood in terms of the mechanisms underlying the differences between AHG and DHG, as well as the variability among different systems. In this study, we attempt to address these basic issues. Two hypotheses are proposed: (a) the different geomorphic relationships represented by AHG and DHG result from the control of lateral adjustment of the bank and flow turbulence over short and long timescales, respectively; and (b) the systematic variability of the AHG and DHG exponents is related to the description of the frictional resistance. These two hypotheses are embedded in our theoretical models and lead to explicit functional forms for AHG and DHG. The verification of our hypotheses is based on a large data set consisting of over 550 b-f-m exponents and 120 power law hydraulic relations. The analysis highlights the role of uncertainties in data acquisition and theoretical/statistical explanations. In addition, the theoretical expressions of AHG also provide an explanation of at-many-stations hydraulic geometry (AMHG) in a physical sense. Overall, our work provides new insights into the fundamental theory of power laws and hydraulic geometry.

中文翻译：

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使用基于过程的模型合理化水力关系之间的差异

用幂律形式描述水力几何是一门具有70多年历史的经典课题。已经提出了两种不同形式的幂律：站内水力几何学 (AHG) 和下游水力几何学 (DHG)。尽管这些半经验表达式的效用已得到广泛认可，但就 AHG 和 DHG 之间差异的潜在机制以及不同系统之间的可变性而言，它们仍然知之甚少。在这项研究中，我们试图解决这些基本问题。提出了两个假设： (a) AHG 和DHG 所代表的不同地貌关系分别来自于对岸边横向调整和短时间尺度上的流动湍流的控制；(b) AHG 和 DHG 指数的系统变异性与摩擦阻力的描述有关。这两个假设嵌入在我们的理论模型中，并导致 AHG 和 DHG 的显式函数形式。我们假设的验证基于一个包含超过 550 个bfm指数和 120 幂律水力关系。分析强调了不确定性在数据采集和理论/统计解释中的作用。此外，AHG 的理论表达式还提供了物理意义上的多站水力几何（AMHG）解释。总的来说，我们的工作为幂律和水力几何的基本理论提供了新的见解。