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 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 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 , alluvial meanders, bedrock meanders, and numerical calculations

1. INTRODUCTION are formed when channel’s 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 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 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 , 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

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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 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 (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 fora  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  q1  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 n R   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 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 takes place on a 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 from bottom to top of the bank as achieves the same height as the 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 tanBc  (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. Also, Fig.5 Case1(b) shows the areal frac- is very complicated, and it takes long time to tion of alluvial cover in the channel. The bedrock compute, so it is not treated in this study. So we channel is fully covered with deposits. The defined a new parameter fland which indicates a result initial alluvial cover was decreased in Case 2 and as of land accretion progress. If fland is equal to 1, this shown in Fig.5 Case2(a) erosion was observed in means land accretion is occurring in all of the rarely outer as well inner bank. Thin initial alluvi- submerged area. If fland is equal to 0, this means that no land accretion occurred. When the water depth Table 1. Numerical conditions goes below minimum depth (hmin) and continues to stay below minimum water depth, the area is treated Grain diameter size 0.74mm as dry. We cut the inner bank calculated as multiple Wavelength 100cm of fland with width of dry area. As it is difficult to Slope 0.01 define fland, in this calculation fland is set as 0.3 which 3 15) Water discharge 0.0005m /s in accordance with Asahi’s study . Meander angle 60 degree Bank Height 10cm 3. NUMERICAL CONDITIONS Table 2. Sediment cover thickness In this paper, we performed calculations to compare characteristics of alluvial and bedrock Case Initial Sediment Sediment supply meanders. Initial channel width is 5cm. Bank height Cover (meter) condition (m2/s) is 10 cm. Initial grid resolutions is 4.2cm (longitu- Case 1 0.01 0.6*10-5 -5 dinal direction) x 5 cm (transverse direction). BR and Case 2 0.005 0.6*10 BL are same with transverse grid size. The hydraulic Case 3 0.005 0.1*10-5 roughness is set as 2.5 times of grain size. The βbank used in this study is 2.5, βbed =1. All the numerical conditions are explicitly mentioned in Table 1. These are almost same with the conditions we used and succeeded to reproduce laboratory exper- iment of bedrock meander16). And we also succeeded to reproduce an alluvial meander in sine-generated curve channel17). In this study, we perform a nu- merical experiment under the condition that validity of the model is confirmed to some extent.

4. NUMERICAL CALCULATIONS, RESULTS AND DISCUSSIONS

(1) Effect of alluvial cover on bedrock meander In order to observe how sediment cover effects the shape of bedrock meander, we performed 3 calculations using similar numerical conditions as mentioned in Table 1. The initial sediment cover thickness and sediment supply were varied in each case. A detail of sediment cover condition and sediment supply is provided in Table 2. The sedi- ment cover decreases from Case 1 to Case 3. The water discharge, slope, meander angle, grain diam- eter size was used as mentioned in Table 1. Each simulation was run for 45,000 seconds. We compared the water depth and alluvial cover of each case. We observed that, when initial alluvial cover is higher, as in Case 1, the outer banks of Flow bedrock are eroded. Erosion of outer banks is shown Fig. 5 Effect of sediment on bedrock meanders. Image (a) of explicitly in Fig.5 Case1(a). Fig.5 Case1(a) also each case shows the water depth and (b) shows the areal frac- shows the migration of bedrock meander from initial tion of alluvial cover

I_838 al-thickness limit the sediment transport, especially performed calculations for both bedrock and alluvial in downstream section, thin point bar formation meanders. Similar conditions as mentioned in Table takes place in the inner bend, making inner bend 1 were used to perform the calculations. Since we prone to erosion. Also, the migration of bedrock want to compare the effect of the differences be- meander was almost negligible towards the down- tween bank materials of bedrock and alluvial me- stream of the channel. Due to a thin point bar for- anders, all other hydraulic conditions must be simi- mation, sediment cannot flow towards the outer lar. We require a condition in which, even if the river bend, which eventually decreases the erosion in bed height changes, bedrock of the river bed should outer bends and hence the migration of channel is not be exposed. Bedrock is exposed only near the limited. In upstream section, enough sediment sup- banks of the channel. Hence, initial bed alluvial ply is given from upstream end, thick point bar is thickness is infinity in all cases. The sediment supply formed, making outer bend prone to erosion as condition is in dynamic equilibrium in which sedi- shown in Fig.5 Case2(b). In Case 3, the sediment ment is supplied so as not to change the height of cover and supply both are decreased, Fig.5 Case3(a) riverbed at upstream end. The amount of the sedi- distinctly shows that there is no outer bank erosion in ment that is transported out, the same amount of this case. Also, the meander doesn’t migrate. Bed- sediment is introduced into the upstream end. rock is only partially covered with alluvium as shown in Fig.5 Case3(b). This happens because no (2) Comparing bedrock and alluvial meanders our very thin point bar formation takes place when In order to reproduce the differences in alluvial sediment supply is extremely low. As a result, flow and bedrock meanders, we performed three calcula- is not sinuous making the flow and sediment hit the tions. inner bank and eventually eroding the inner bank. In Fig.6(a) it can be seen that the tilt in bend apex These simulations provide a cogent evidence for of alluvial bend is towards the upstream of the the importance of sediment for formation of bedrock channel. This kind of meandering is famously meanders. As, the bedrock migration is more likely known as kinoshita meander. Whereas, Fig.6(b) during higher sediment availability, we made an shows the meandering migration in bedrock channel. attempt to perform calculations with dynamic equi- It is evident that the tilt in bend apex in bedrock bend librium sediment supply and thick alluvial cover. We is towards the downstream of the channel. Fig.6(a)

Fig.6 Meander bend migration. (a) Alluvial meander without bank accumulation (b) Bedrock meander without bank accumulation. (c) Alluvial meander with land accretion

I_839 does not provide a very clear appearance of tilt in the 3) Kinoshita, R. & Miwa, H.: River channel formation which bend apex towards the upstream of alluvial channel. prevents downstream translation of transverse bars (in Japanese) Shinsabo,Vol. 94, pp.12–17, 1974. As this calculation inhibits inner bank accumulation, 4) Lawrence,S.: Fluvial Hydraulics, (Google books) 2009. the calculations cannot be carried on for a longer 5) Asahi, K., Shimizu Y., Nelson J., and Parker G. : Numer- duration of time because the calculations will fail ical simulation of river meandering with self-evolving due to grid shape strain caused by excessive bank banks, J. Geophysical Research:Earth Surface, Vol. 118, erosion. Hence, in order to get much distinctive pp. 2208-2229, 2013. 6) Parker, G., Shimizu Y., Wilkerson G.V., Eke E.C., Abad results for kinoshita alluvial meanders, we per- J.D, Lauer J.W., Paola C., Dietrich W.E. and Voller V.R.: formed calculations with active bank accumulation A new framework for modeling the migration of mean- system. Fig.6(c) indicates explicitly, the tilt of bend dering rivers, Earth Surf. Processes Lanforms, Vol. 36, apex towards the upstream of alluvial channel. Sim- issue 1, pp. 70-86, 2011. ilar behavior of alluvial and bedrock meanders were 7) Meshkova, L. V. and Carling, P. A.: Discrimination of alluvial and mixed bedrock–alluvial multichannel river observed in real rivers, showed in aerial photographs networks, Earth Surf. Process. Landforms, Vol. 38, pp. of Atlanta river and Shikaribetsu river in Fig. 1 and 1299–1316, 2013. Fig. 2 respectively. 8) Inoue, T., Izumi, N., Shimizu, Y., and Parker, G. : Interac- Seminara18) employed a linear model of flow and tion among alluvial cover, bed roughness, and incision rate bed topography. They showed that meanders behave in purely bedrock and alluvial-bedrock channel, J. Geophys. Res. Earth Surf., 119, 2014. as linear oscillators. They resonate at some values of 9) Inoue, T., Iwasaki, T., Parker, G., Shimizu, Y., Izumi, N., aspect ratio of channel and meander wavenumber. In Stark, C., and Funaki, J. : Numerical simulation of effects this study, he suggested that downstream migration of sediment supply on bedrock channel morphology, J. and upstream skewing (the tilt of bend apex towards Hydraul. Eng., 2016. the upstream) of meander patterns is obtained under 10) Chatanantavet, P. and Parker G.: Physically based model- ing of bedrock incision by abrasion, plucking, and macro- sub-resonant conditions. Their results were in fair abrasion, J. Geophys. Res., Vol. 114, pp. F04018, 2009. 3) agreement with Kinoshita’s laboratory observa- 11) Sklar, L. S., and Dietrich W. E.: A mechanistic model for tions. And our results of alluvial meanders are con- into bedrock by saltating bedload, Water sistent with his study. However, our results of bed- Resources Research, Vol. 40, pp. W06301, 2004. rock meanders show downstream migration with 12) Inoue, T: Numerical simulation of a bedrock-alluvial river bend that cuts downward and migrates laterally, both via downstream skewing (the tilt of bend apex towards incision, Gravel bed river 8, 2015. the downstream). This may be an interesting phe- 13) Mishra J., Inoue T., and Shimizu Y.: Effect of bank erosion nomenon in a mixed bedrock-alluvial meandering on bedrock , IAHR APD, 2016 channel. 14) Parker, G., Shimizu Y., Wilkerson G.V., Eke E.C., Abad J.D., Lauer J.W., Paola C., Dietrich W.E. and Voller V.R. A new framework for modeling the migration of mean- 5. CONCLUSIONS dering rivers, Earth Surf. Processes Lanforms, vol. 36, is- sue 1, pp. 70-86, 2011. In this study, we used numerical simulations to 15) Asahi, K.: Numerical simulation of river meandering with first prove that bedrock meanders require sufficient self-evolving banks. Thesis submitted to Hokkaido Uni- versity. (2014) alluvial cover and sediment supply for its formation. 16) Jagriti M., Inoue T., and Shimizu Y. : Simulations of Lat- We also successfully reproduced the characteris- eral Erosion in Bedrock Channels, Abstracts of Applied tics of alluvial and bedrock meanders observed often Mechanics, 100077, 2016. in field. Alluvial meander’s bend tends to tilt in the 17) Jagriti M., Asahi K., Shimizu Y., and Parker G.. Numerical direction of upstream which is in contrast to Bedrock investigation of Channel evolution considering bank ero- sion and land accretion, RCEM, 2015. meanders, in which the bend tilts towards the 18) Seminara, G. : Meanders, J. Fluid Mech, Vol. 554, pp. 271– downstream of the channel. 297,2006.

ACKNOWLEDGMENT: The first author of this paper is thankful to Japanese Government (MEXT) (Received September 30, 2016) for aiding this research. Participation of T. Inoue in this research was made possible in part by JSPS KAKENHI Grant Number 15K18126.

REFERENCES 1) Parker, G., Diplas, P and Akiyama, J.: Meander bends of high amplitude, J. Hydraulic Eng., Vol. 109, Issue 10, pp. 1323-1337, 1983. 2) Parker, G. and Andrews, E.D. : On the time development of meanders bends, J. Fluid Mech., Vol. 162, pp. 139-156, 1986

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