Journal of Japan Society of Civil Engineers, Ser. B1 (Hydraulic Engineering), Vol. 73, No. 4, I_835-I_840, 2017. C OMPARISON OF BEDROCK AND ALLUVIAL MEANDERS USING 2D MODELLING Jagriti MISHRA1, Takuya INOUE2 and Yasuyuki SHIMIZU3 1Member of JSCE, PhD Student, Department of Field Engineering for the Environment, Hokkaido University (Sapporo 060-8628, Kita-ku, Kita-13, Nishi-8, Japan) E-mail:[email protected] 2 Member of JSCE, Dr. of Eng., Researcher, Civil Engineering Research Institute for Cold Region (Sapporo 062-8602, 1-jo 3-choume Hiragishi, Toyohira-ku, Sapporo, Hokkaido, Japan) E-mail: [email protected] 3Member of JSCE, Dr. of Eng., Professor, Department of Field Engineering for the Environment, Hokkaido University (Sapporo 060-8628, Kita-ku, Kita-13, Nishi-8, Japan) E-mail: [email protected] Understanding the differences between characteristics of alluvial and bedrock meanders is the need of hour. In this paper we have performed some numerical calculations in order to, first make an effort to observe the change in migration of bedrock channel in response to change in sediment cover and sediment feed rate. After realizing how sediment availability effects bedrock meanders, we tried to compare characteristics of alluvial and bedrock meanders under similar hydraulic and physical conditions. Key Words : meandering rivers, alluvial meanders, bedrock meanders, and numerical calculations 1. INTRODUCTION are formed when channel’s sediment transport ca- pacity exceeds channel’s sediment supply4). The Almost all rivers tend to follow a sinusoidal characteristics of meandering in alluvial and bedrock shape as they move. This sinusoidal shape is deter- channels are noticeably different from one another, mined by various factors, with the key factors being- the difference in tilt direction of bend is one such climate and discharge conditions, sediment load, example. local tectonics and rock strength of channel. Mean- Various attempts have been made in the past to dering is not just restricted to alluvial rivers, it is a understand alluvial rivers5), 6). Despite various efforts very common phenomena in bedrock rivers as well. to explore alluvial rivers, Kinoshita type meandering A very well-known meandering shape, often called still lacks literature. Also, bedrock meanders haven’t as “Kinoshita type meandering” was first observed been studied and explored widely. A few attempts in Ishikari river of Japan. Kinoshita type meanders have been made to identify ways to distinguish be- are asymmetric unlike sine generated curves. They tween alluvial and bedrock rivers7). A broader un- are characterized by having multiple point bar derstanding of differences in characteristics of allu- growths in the inner bank of large-amplitude me- vial and bedrock meanders is needed. ander bends. Kinoshita type meanders are a common In this paper, we have made an attempt to reason sight in alluvial rivers with high curvature and high why this difference in the tilt of bend occurs. We amplitude bends1), 2), 3). Fig.1 provides an insight to have used 2 Dimensional model to simulate this the shape of kinoshita meander, as marked by the behavior of alluvial as well as bedrock rivers. We square box, kinoshita meander’s bend tilts towards made an effort to understand the effect of bank ero- the upstream. sion on the shape of meander. We have also calcu- During a field visit to a bedrock river, Shikari- lated how alluvial cover thickness and supply rate of betsu in Hokkaido region of Japan, we observed that sediment can affect the formation of meanders in the tilt of bend was towards the downstream of the bedrock channels. upstream of river as shown in Fig.2. Bedrock rivers I_835 The tilt of bend apex is towards the Upstream of river The tilt of bend apex is towards the Upstream of river Fig.1 Kinoshita type meandering in Atlanta river, Alaska, USA. Image courtesy: Google Images The tilt of bend apex is towards the Downstream of river Fig.2 Meandering Shikaribetsu river. It is a bedrock river in Hokkaido, Japan. Image courtesy: Takuya Inoue 2. NUMERICAL MODEL where ηb is the elevation of bedrock layer, βbed is the abrasion coefficient of the bedrock-bed, qbs is the A 2-dimensional plane flow state is simulated by sediment transport rate per unit width in streamwise the basic equations of the numerical model for allu- direction s, and qbn is the sediment transport rate per vial meanders presented by Asahi et al.5). Alluvi- unit width in transverse direction n. In bedrock al-layer deformation and sediment transport on channel, the sediment transport rate is estimated by bedrock channel are simulated by the system pre- adjusting the sediment transport capacity in ac- sented by Inoue et al. 8), 9). Since the detailed de- cordance with the amount of gravel present on the scriptions are described in the referenced papers, we bedrock8). When the amount of gravel is zero, the do not discuss it in detail in this paper. sediment transport rate is zero. When the amount of In this section, we mention the governing equa- gravel is enough, the sediment transport rate is equal tions for bedrock bed erosion (i.e., vertical erosion), to the sediment transport capacity. Following equa- bedrock bank erosion; (i.e., lateral erosion), alluvial tion is used for the areal fraction of alluvial cover pc. bank erosion and land accretion due to inner bank accumulation. In numerical calculation system, the aaL for0 L pc (2) governing equations are changed into moving 1 fora L boundary-fitted coordinate system5), but we describe the equations in a curvilinear coordinate system for simplicity. where ηa is thickness of alluvial layer, L is bedrock macro roughness. ηa is calculated by sediment con- (1) Bedrock bed erosion tinuity equation in the same way as in the general The erosion of bedrock bed is calculated by fol- model for alluvial deformation. When the bedrock lowing equation9). surface is smooth, L is roughly 2 times of gravel diameter d. b 22 q q1 pc (1) (2) Bedrock bank erosion t bed bs bn The saltating gravel particles attack riverbed, I_836 hence magnitude of bedload transport rate, controls erosion rate of bedrock bed11).However, the erosion Cross-sectional profile of q rate of bedrock bank may not depend on the mag- bn nitude of bedload transport rate12). For example, n when gravel moves parallel to bedrock sidewall, the number of gravel colliding with sidewall is theoret- Lbank Lbank ically zero. The experimental result of Jagriti et al. 13) showed that the bedrock bank erosion rate of straight B channel is almost zero. Inoue assumed that the ero- Fig.3 Definition of Lbank sion rate of bedrock bank depends on lateral bedload 12) transport rate qbn 0 nR q (3) bank bn n n0 L t R bank B 0 nL q (4) bank bn n n0 L t L bank where βbank is the abrasion coefficient of the bedrock bank, Lbank is an estimate of the distance of the boundary layer over which the transverse bedload Fig.4 Variables used in river bank erosion model. rate decreases to zero at the bank, as shown explic- (Courtesy: Asahi, 2014; thesis submitted to Hok- 0 0 itly in Fig.3. nR , nL are axis values for both banks as kaido University) shown in Fig.4. 0 (3) Alluvial bank erosion n 1 Z 0 1 L BL This numerical model was implemented by Asahi t 1 5) t tan et al considering a framework for modelling the Bc (8) migration of meandering rivers which was proposed qbs 1 14) qbn nn 0 by Parker et al . In a curvilinear coordinate system sBL L in Fig.4, the height of arbitrary position on the bank of alluvial river can be shown as where, λ is porosity of bank material, BR and BL are the widths of right bank and left bank, respectively. 0 0 (5) right bank: Z BR Z BR n nR tan θBc (4) Land accretion 0 0 Land accretion affects the shape of river over left bank: Z Z n n tan θ (6) BL BL L Bc long time. Sediment deposition takes place on a point bar located at the convex part of the river. As 0 0 time proceeds this area becomes higher. As this area where ZBR , ZBL are the height of bottom of bank, respectively. That is, these are the bed height at the is higher than channel, it rarely submerges. 0 0 Vegetation can grow in this area. Vegetation works connection point between bank and bed. nR , nL are 0 0 as a resistance to the stream, and catches fine the axis values for ZBR , ZBL in n direction. θBc is the bank collapse angle. Alluvial bank shifting caused sediment during flood, as a result the area's elevation by bank erosion can be obtained by integrating grows up higher than before. With time, this area Exner equation from bottom to top of the bank as achieves the same height as the floodplain and it 5) mentioned below: submerges only during floods . In estimating the amount of bank shift by land 0 accretion, it is important to specify the area that n 1 Z 0 1 rarely submerges. To specify this area, discharge, R BR t 1 which indicates ordinary flow, was used in this t tanBc (7) model because it is hard to specify the area during qbs 1 flood. However, it is hard to assume that land q 0 accretion takes place at all of the specified area, as its sBbn nn R R progress depends on various factor (ex: sediment type, vegetation type, and climate). Relationship I_837 between those factors and land accretion phenomena channel.