Scaling Behavior of Gravel Surfaces
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Scaling Behavior of Gravel Surfaces Jie Qin · Deyu Zhong · Sai Leung Ng · Guangqian Wang
Received: 25 July 2011 / Accepted: 6 February 2012 / Published online: 2 March 2012 © International Association for Mathematical Geosciences 2012
Abstract Roughness of successively developed gravel surfaces in flume experiments is investigated using laser scanned elevation fields. Scaling behavior of these surfaces is studied using structure functions. The results show that all surfaces exhibit good scaling behavior for two scaling regions. Multifractal analysis based on singular measures is then conducted for the two scaling regions, respectively. The relatively smaller scaling region characterizing subgrain scale roughness exhibits evident nonstationarity and intermittency. In contrary, the measures for the larger scaling region characterizing the spatial distribution of grains are more stationary and less intermittent than the measures for the subgrain scaling region. Keywords Multifractal analysis · Scaling invariance · Armouring process · Roughness
1 Introduction Bed roughness is a fundamental issue for river hydraulics and fluvial geomorphology which influences properties of the average flow, turbulence, flow resistance, J. Qin () · D. Zhong · G. Wang State Key Laboratory of Hydroscience and Engineering, Tsinghua University, Beijing, China e-mail: [email protected] D. Zhong e-mail: [email protected] G. Wang e-mail: [email protected] S.L. Ng Department of Geography and Resources Management, The Chinese University of Hong Kong, Hong Kong, China e-mail: [email protected]
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Math Geosci (2012) 44:583–594
and bed particle motion (Nikora et al. 1998; Aberle and Nikora 2006). Traditionally, the roughness of gravel beds has been represented by a characteristic particle size of bed material, dx . However, due to different properties of individual particles and the spatial arrangement of them, the bed roughness cannot be adequately described by a single grain size. A more straightforward way to describe the bed roughness is directly characterizing the elevation field. Given that roughness is scale dependent, it is sensible to derive roughness characteristics which are independent of scales and fractal analysis may provide particularly fruitful for this purpose (Butler et al. 2001). Second-order structure function has been extensively employed to analyze fractal characteristics of gravel beds (Robert 1988, 1991; Furbish 1987; Marion et al. 2003; Cooper and Tait 2009). However, water-worked gravel beds formed in both flume experiments and field conditions often demonstrate deviations from the normal distribution such that higher-order statistics should be considered for the description of roughness (Marion et al. 2003; Smart et al. 2004; Nikora and Walsh 2004). Nikora and Walsh (2004) and Aberle and Nikora (2006) applied high-order structure functions for studying water-worked gravel surfaces measured at field sites and a large laboratory flume. The analysis proved that both field and experimental gravel surface
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