A Statistical Analysis of Bank Erosion and Channel Migration in Western Canada

A statistical analysis of bank erosion and channel migration in western Canada

GERALD C. NANSON Department of Geography, The University of Wollongong, P.O. Box 1144, Wollongong, New South Wales, Australia 2500 EDWARD J. HICKIN Department of Geography, Simon Fraser University, Burnaby, B.C. Canada V5A 1S6

ABSTRACT (co), channel width (fV), the force per unit area of the outer (concave) bank which resists channel migration (rj,), the bank height (h), and the

Mean lateral-migration rates for 18 meandering river channels in bend radius of curvature (r). Further, we have shown that Tb is largely a western Canada are explained statistically in terms of hydraulic and function of the size of sediment at the base of the channel (D¡Q), such that sedimentological variables. The volume of sediment eroded from the outer bank of a meander bend is shown to be largely a function of M= f (co, W, 7)50, h, r). (1) river size and grain size of sediment at the base of the outer bank. These variables explain almost 70% of the volumetric migration rate Sediment load (particularly bedload) is known to be strongly corre- for these relatively large, sand- and gravel-bed streams. It would ap- lated to channel migration rate (Neil, 1984), but whether the relationship pear that bank erosion and channel migration are essentially problems is causal or dependent has not been clearly demonstrated. Almost no of sediment entrainment which is dependent on total stream power sediment load data exist for the 18 rivers chosen; therefore, the role of and sediment size. Vegetation on the outer bank is seen to have little sediment load cannot be independently evaluated. Bagnold (1980), how- significant effect in controlling channel migration. Further refinements ever, has shown that bedload transport is largely a function of stream of the type of data used here should permit the development of an power operating on particular sediment sizes, and, as both of these varia- accurate predictive model of regional channel migration. To this ef- bles are included in equation 1, a bedload transport term is implicit. fect, it is most important to develop a precise relationship between Previous researchers have examined bank erosion and channel migra- bank resistance and the size of sediment at the base of the outer bank. tion from a number of aspects. Planform of meander bends has long been considered an important variable. Leighly (1936) showed how changes in INTRODUCTION the position of the high-velocity filament in a bend, from low to high stage, could provide a mechanism to account for the evolution of meander bends Channel migration and associated river-bank erosion are among the through a series of predictable planforms. Daniel (1971) demonstrated most dynamic geomorphological processes and, therefore, of considerable that channel length around a meander loop increases in proportion to the scientific interest. Furthermore, man's intensive use of river valleys and magnitude of the channel-forming discharge, whereas Hickin (1974) dem- floodplains means that a detailed understanding of these processes is of onstrated that migration operates to maintain a minimum curvature ratio considerable economic value and engineering significance. (bend radius to channel width: r/W) of slightly >2. From dendrochrono- Numerous descriptions of particular cases of channel migration exist, logical evidence, Hickin and Nanson (1975) showed that bend migration but, for the most part, these are restricted to a few bends on one or two reaches a maximum value as the curvature ratio approaches 3 and declines channels (see Table IV in Hooke, 1980). A lack of accompanying basic rapidly on either side of this ratio value. Indeed, Carey (1969) and Page hydraulic, geomorphologic, and sedimentologic data makes it impossible and Nanson (1982) have shown that, in very tightly curving bends, deposi- to combine more than a few of these observations in order to derive a tion will occur around the outer bank and erosion will occur at the convex general model of channel migration for a range of environmental condi- bank. tions. Indeed, there appears to have been no attempt to systematically The intermittent nature of channel migration has been demonstrated examine a range of migration rates for a variety of river types. The objec- by Brice (1973), who found that meander loops along the same reach of tive of this paper is statistically to relate bend migration rates to channel the White River do not evolve sequentially or simultaneously, but inter- hydraulic, geomorphologic, and sedimentologic characteristics for 18 mittently, first in one part of the reach and then in another. Hickin (1974) single-thread, meandering river reaches in western Canada in an attempt to obtained similar findings, and Nanson and Hickin (1983) showed that develop a predictive, empirical model of lateral migration. In the strictest migration can be very discontinuous in time as well as in distance along a sense, this model will have direct application only for rivers in physio- single reach. Bends can remain stationary for tens of years, thereby making graphic and climatic settings very similar to those in western Canada. difficult the estimate of long-term migration rates from short-term More importantly, at this early stage of investigation into channel migra- measurements. tion, however, these empirical results identify at a general level those A rather different approach to the problem of channel migration has variables that are most influential in determining river migration rate. been developed by those focusing on the details of bank erosion without In earlier work (Hickin and Nanson, 1984), we suggested that the specific regard to channel planform. The role of frost action and ground ice rate of channel migration (M) is likely to be dependent on stream power has been considered by Wolman (1959), Walker and Arnborg (1966), (essentially, the product of discharge and slope) per unit area of the bed and Outhet (1974). Knighton (1973) found that bank erosion at a cross

Geological Society of America Bulletin, v. 97, p. 497-504, 5 figs., 3 tables, April 1986.

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section was largely determined by the magnitude and variability of dis- Despite these numerous studies of bank erosion and channel imita- charge and by the degree of asymmetry in the velocity field, bank wetting tion, the problem remains largely unquantified. Only Hooke (1979,1980) being a particularly important preconditioning process. More recent work has attempted to develop predictive statistical relationships. Most of the has shown the importance of basal sediments in composite banks (non- ten variables that she assessed, however, were not sufficiently independent cohesive sand and gravel overlain by cohesive sandy silt) in cases wherein of one another to allow unambiguous results, and they were selected from cantilever collapse brings down the cohesive overburden (Thorne and a very restricted range of river environments. Hooke, however, did find, as Tovey, 1981). Impoi.tantly, this work recognized that removal of both the did Daniel (1971), that erosion rate is related to catchment area (dis- basal sediment and the collapsed blocks is dependent on fluvial processes charge) and the percentage of silt and clay in the banks. although the predominant failure mechanism is not directly fluvial (see Most of the above work on bank erosion has been for relatively also Thorne and Lewin, 1979). Their work also acknowledges the role of low-energy streams, but, because the erosion-resistance thresholds are) re- bank vegetation in limiting erosion, a problem examined quantitatively by lated particularly to bank vegetation and fine-sediment cohesion, these Smith (1976). observations may not compare closely to similar processes operating on

STUDY RIVERS

1 Little Smokey Lake 2 Milk 'Athaba s ca 3 Belly 4 West Prairie 5 Beaver 6 Waterton 7 Eagle (Upper) 8 Eagle (Lower) 9 Swan 10 Shuswap (Upper) 11 Pembina 12 Muskwa 13 Oldman 14 Shuswap CLower) 15 Chlnchaga 16 prophet 17 Slkannl Chief 18 Ft. Nelson'

km PACIFIC OCEAN

Figure 1. Locations of the study sites in western Canada.

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TABLE 1. CHANNEL CHARACTERISTICS FOR THE 18 STUDY REACHES

General channel character Special features Basal sediments of outer bank

Channels in which Flood-plain sediments Beaver Banks are free of vegetation Noncohesive silty fine-medium sand sand dominates the display typical upward-fining Swan Deep, narrow, canal-like channel; Nonoohesive silty fine sand; lower boundary sequence; some gravel vegetated banks lenses of medium sand 0.18 mm materials lenses appear in West Prairie 4 Deep, narrow, canal-like channel; Noncohesive medium sand lower strata vegetated banks Pembina 11 Bank vegetation variable; scrub grass. Noncohesive medium to coarse sand Artificial cutoffs have resulted in some DJQ 0.5 mm accelerated erosion Chinchaga Herbaceous vegetation on outer banks Medium sand; no silt Dtn 0.35 mm Eagle(L) Deep canal-like channel; banks well Medium to coarse sand D$Q 0.72 mm

Milk 2 Little bank vegetation; cohesive silt Well-sorted sand with lenses of silt over noncohesive sand and fine gravel £>

Transitional channels: Well-developed sand and Sikanni Chief 17 Log jams are common; no bank vegetation Medium sand to fine gravel; 5 mm

Sand and gravel gravel point bars. isolated gravel Z>5Q 60-120 mm mixture in the lower Flood plains are silt over Shuswap (L) 14 Cutbanks fine sandy silt over sandy Medium sand to fine gravel, D$Q 15 mm boundary materials sand and fine gravel gravel. Point bars fine gravel & sand Belly 3 Root mat strengthens banks in forested Gravel D^Q 12 mm and some coarse sand areas

Channels in which Flood-plain sediments may Shuswap(U) Banks of silty sand over medium gravel Gravel Z?JQ 15 mm gravel dominates the be silt but basal Waterton Gravel lag deposits near base; some Gravel 45 mm; no sand, some gravel lower boundary sediment is coarse; root rip rap; silty overburden lag, ¿>50 «250 mm materials gravel may be in matrix Roots important binding agent in upper Gravel Z)^ 40-50 mm; no sand of fines bank EagleOJ) Dense root mat in upper bank Gravel D$Q 45 mm; some coarse sand Little Smokey Some very large boulders present; strong Large cobbles «250 mm; no sand root mat in upper bank Some lag gravel Muskwa Considerable gravel stored in channel bars Gravel 90 mm; no sand Prophet Cutbanks fine sand & silt over coarse Gravel £>50 80-100 mm; coarse sand gravel and sand; no bank vegetation

large rivers. This study attempts to develop relationships between channel transparencies of British Columbian, Albertan, and Canadian Government migration rates (bank erosion) and generalized flow and channel charac- 1:15,000 to 1:40,000 aerial photographs. Channel displacement was mea- teristics for medium- to high-energy rivers in order to, first, generate a sured by superimposing two images of the same channel reach time-lapsed predictive empirical model and, second, assess the relative importance of by 21 to 33 yr. Common registration points were used to match each pair each of the measured variables as to their influence on migration. of images, using a projector equipped with adjustable focal length and triaxial mount. Channel migration rate for a bend was measured as the THE STUDY SITES maximum outer-bank displacement normal to the former channel axis (Hickin, 1974). The 18 river reaches examined here are located in British Columbia Channel width for each bend was taken as the distance between and Alberta (Fig. 1). The reaches are of single-thread, meandering, gravel- opposite bank crests viewed from aerial photographs. These distances were and sand-bed rivers for which sequential aerial photography provides averaged for several measurements near points of inflection and along evidence of channel shifting during periods of 21-33 yr. These particular straight reaches at a spacing of approximately one channel width along the reaches were chosen because they represent a wide range of river and channel center line. Radius of channel curvature was measured about the sediment sizes. Discharges at the 5-yr annual-series flood vary from 44 to bend axis of the modern channel trace in accordance with the procedure 3,972 m3 s_1; channel slopes, from 0.0016 to 0.00022; sediments, from described by Nanson and Hickin (1983). Both width and bend-curvature fine sand to large cobbles; and migration rates, from 0 to 12.6 m/yr. All measurements represent conditions at bankfull flow. Bank height, the ele- sites were free from land use disturbance and were easily accessible. The vation difference between the channel thalweg and the top of the adjacent bends measured for migration were, as could best be determined from outer bank, was measured at several points around a number of bends and aerial photographs, freely meandering within floodplain alluvium. A brief then averaged for each reach. description of each study reach is given in Table 1. The 5-yr floods on the annual series were calculated from Water Survey of Canada records for 14 rivers; values for the remaining 5 rivers DATA DETERMINATION that did not have flow records were based on ground surveys of channel cross section with application of the Manning formula and regional hy- Individual measurements of channel migration, channel width, and draulic geometry relations (Bray, 1975,1979). Our field experience in the radius of channel curvature were obtained for 118 bends on 18 river Beatton River and Fort Nelson River basins suggests that there the 5-yr reaches. In addition, average slope (S) height of the concave, outer bank flood on the annual series approximates bankfull flow, and this relation- (A), median diameter of basal sediments in the outer bank (£50), and ship is assumed to apply to the 5 channels in question. magnitude of the 5-yr flood (2s.o) were obtained for each study reach The appropriate slope parameter in this study should be the mean using methods detailed below (Table 2). water surface slope at the 5-yr flood, but these data were not available. Channel planform data were measured from maps drawn at scales of Instead, slopes from 1:25,000 maps were measured, low-stage water- 1:500 to 1:1,500. The maps were prepared from screen images of 35-mm surface slopes were surveyed for most of the rivers, and the two were

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TABLE 2. HYDRAULIC, SEDIMENTOLOGfC, AND GEOMORPHIC DATA

River No. S n W k r M' N °50 b (mV) (m km"1) (Wnr1) (m) (m) (my"1) (mm) (Nm"

Little Smokey 1 44 1.70 734 37 2.5 0.57 8 90.00 515 Milk 78 0.92 705 48 3.5 1.68 4 1.40 120 Belly 101 1.50 1488 40 2.5 1.18 7 12.00 504 West Prairie 102 0.69 1037 30 6.5 0.86 3 0.37 186 Beaver ;; 244 0.28 671 52 4.0 1.41 10 0.18 119 Waterton (1 247 1.90 4609 70 2.8 3.93 3 45.00 419 Upper Eagle V 272 1.09 2912 49 3.5 1.34 5 45.00 621 Lower Eagle ¡; 272 0.29 773 50 5.0 0.71 4 0.72 218 Swan 0 275 0.32 855 46 6.5 1.52 5 0.18 87 Upper Shuswap 10 306 1.00 3005 63 2.0 1.73 2 17.50 868 Pembina 11 369 0.22 795 79 5.5 2.60 12 0.50 56 Muskwa 1:: 377 1.51 5590 49 4.0 2.65 8 90.00 527 Oldman 1:1 383 1.60 6005 93 3.0 7.26 4 45.00 276 Lower Shuswap 454 0.27 1203 92 2.5 1.89 3 12.00 255 Chinchaga 15 766 0.25 1881 90 6.0 1.03 23 0.37 304 Prophet n; 830 1.60 13014 140 6.0 2.34 5 90.00 927 Sikanni Chief r 1259 0.45 5552 127 8.3 2.92 3 6.00 229 Fort Nelson in 3972 0.29 11288 278 7.0 4.44 9 1.40 363

found to be in good agreement. Because channel migration rates were migration rate occurs at 2.0 < r/W< 3.0 (Fig. 3). The vertical scatter of determined for reaches of river far greater in length than could be mea- points within the envelope occurs, in part, because they are from a family sured for field slopes, the more extensive map slopes have been used where of curves, one for each of the 18 different reaches. The distribution is possible. In the case of the Lower Eagle, Fontas, Sikanni Chief, and Lower similar to the theoretical distributions shown in Nanson and Hickin (1S'83) Shuswap Rivers, however, a lack of detailed map contours meant that (Fig. 4). only field slopes wers usable. The relationships between migration rate and the variables in equa- The sizes of the basal sediments near the low-water line in the outer tion 1 are difficult to isolate because of the complex relationship between (concave) banks were roughly measured in the field, using a ruler and migration rate and bend curvature (Figs. 2 and 3). Insufficient data v/ere fine gradicule at a number of locations in each reach. On the basis of these obtained to define an accurate curvilinear relationship for each of the 18 observations, each reach was then assigned an appropriate Wentworth river reaches. In an attempt to resolve this problem, a subset of bends was textural class and associated median diameter (£>50) for the statistical selected for 2.0 < r/W< 4.0, thus holding curvature essentially constant. analysis to follow. Eioth the considerable length of each study reach and Because most of the measurements cluster within this range, relatively few the heterogeneous character of much of the basal sediment act to diminish (22%) of the collected data had to be excluded. This selected subset pro- the relative value of a system of sediment sizing more sophisticated than vided a median migration rate of 1.5 channel widths per century, althcugh that adopted. the distribution is very positively skewed. In ~ 10% of the cases, migration

DATA COLLECTION

Our previous research on the Beatton River has clearly shown that channel migration is a complex function of bend curvature (Hickin and Nanson, 1975; Nanson and Hickin, 1983) (Fig. 2). Similar data presented here for a total of 18 rivers are enveloped by a curve which displays the same basic form as that for the Beatton River data, and a maximum

r/W

r/w Figure 3. The relation between relative migration rate (expressed Figure 2. The relation of migration rate (m/yr) to bend curvature in channel widths per year) and bend curvature ratio (r/W) far all ratio (r/W) for bends on the Beatton River, British Columbia (from field sites. The basic form of the envelope curve is the same as that for Nanson and Hickin, 1983). the Beatton River (superimposed from Fig. 2).

Figure 4. The relation between the coefficient of resistance to lateral erosion (rb; fi/M*Â) and the median diameter of the basal sediments (Z?5o) in the outer bank. Horizontal bars define the Wentworth textural-class representative of the basal grain size in each study reach. River reaches are identified in Table 2.

rates exceeded 5.5 channel widths per century, and there was a maximum rameters) from equation 1, a series of stepwise regressions was executed. rate of ~ 1 channel width per decade. To test for statistical relationshipsamong the variables in equation ^arith- It is also apparent from Figure 3 that during the 21- to 33-yr period of metic means of migration rate (M *) {my'1) and channel width (W) (m) record, -20% of the bends did not move {M/W-0). Some of these cases were calculated for each of the 18 river reaches. These, as well as discharge may be the result of an insufficiently long period of time having elapsed for the 5-yr flood (Qs.o) (mV1), channel slope (5), outer bank height (h)

between photographs for the method to detect what is, by its nature, an (m), and median diameter of the basal sediments in the outer bank (Z)50) intermittent process of channel migration (see detailed evidence of this (mm), were used to derive the parameters of the maximum erosion rate of point in Nanson and Hickin, 1983). If the period of record were longer, the banks. These parameters include volume per unit length of channel at most of these zero values should more closely conform to the long-term the point of maximum lateral displacement per year (M_*h) (m3m~'y_l = mean migration rate. Additional zero values may be the result of certain mV1), power per unit area of the bed (cu = pgQSW'1) (Wm~2), and bends arresting on unknown obstructions within each floodplain. In both total power per unit length of channel (ft = pgQS) (Wm"1) (Table 2). cases, zeros have been interpreted as unrepresentative of the general condi- Because all of these sets of data are strongly positively skewed toward the tion for unconfined meanders during long periods of time, and, therefore, largest rivers, their distributions were normalized by obtaining the log these bends have been omitted from the mean values given in Table 2 and (base 10) of each variable. from the following statistical analyses. Under these conditions, the median The simple correlation matrix of transformed variables (Table 3) migration rate for the data set as a whole is ~2 channel widths per century. indicates that no single measured variable accounts for >50% of the variance in the mean migration rate; total stream power, mean width, and DATA ANALYSIS AND INTERPRETATION discharge provide 48%, 44%, and 34%, respectively. Using stepwise regression, mean migration rate is regressed sequentially against groups of inde- In order to establish the respective variance contributions and hence pendent variables introduced in order of their decreasing simple the predictive value of each independent variable (and their derived pa- correlation. In this procedure, additional independent variables were

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TABLE 3. CORRELATION COEFFICIENTS BETWEEN ALL VARIABLES

M' M'h Q;0 ai W S Dm h il WW ti'h/W W ih (i/U'h

1.00 .85+ .58t ,45t ,66t .13 .27 .07 ,69t .58+ .58t ,60t .01 Afit ,76t .34 ,73t -.17 -.06 .59+ .65+ .30 .73+ .27 -.20

e50 .26 .91+ -.41+ -.07 .53+ ,69t -.23 .21 .48+ .1) «)_' .21 .71+ .70+ -.05 .84+ .36 .29 .24 .71+ W -.27 .08 .37 .70+ -.23 .08 ,70t .1» S .81+ -.52+ .37 .46+ .02 .14 .65+

Dx -.54+ .55+ .27 -.17 ,50t .78+ h .16 -.31 .49+ -.41+ -.39+ Q_ _ .14 .26 .56+ .62+ M_VW_ .67+ .02 -.13 M_*~h/W -.30 -.43+ W/i _ .44+ Cl/M'h 1.00

+ForN= 18 critical value of r (P < 0.1) = 0.38.

added to the relationship only if, in combination, they explained >5% of M*- ^ 0A5a>*m (4) the variance in mean migration and then only if their individual contribu- 20.3 20.3 tion was significant at at least the 10% level. 10.0 10.0 If mean migration rate is regressed against discharge, slope, bank 5.0 height, and median grain size, discharge accounts for 34.1% of the variance and slope accounts for 17.1%. The others have no significant contribution The total erosive work accomplished by the river during the migra- (equation 2). tion of a meander bend is expressed by the annual volumetric erosion rate at the point of maximum bend migration (M *h). Equations 5, 6, £.nd 7 M *= 1.663 g 0.482 £0.368 (2) indicate that the use of Mh clearly increases the predictability of channel 53.2 24.1 17.1 (% explained variance) migration from measurements of channel width, discharge, slope, bank 1.0 1.0 2.5 (% significance of explained height, and basal sediment size. This improvement results because channel variance; F test) migration involves erosion of a volume of material around the outsid; of a 1.0 5.0 (% significance that exponent meander bend, a factor not taken into account if migration is expressed as does not equal zero; t test) a simple linear measure of bend displacement.

If the same regression is attempted, but discharge is replaced by mean M*h = 25.06 g 0.788 s0-74 Dtf 09 (5) width, almost the same level of explanation is achieved, and grain size and 69.1 57.5 2.6 9.0 bank height remain nonsignificant (equation 3). 1.0 1.0 2.5 5.0 1.0 5.0 10.0 M* = 0.301 fp'0.895 ¿0.271 (3) 54.1 43.7 10.4 M*h = 2.089 îy 1.369 £0.568 (6) 1.0 1.0 10.0 62.6 53.6 1.5 7.4 0.1 10.0 1.0 1.0 10.0 10.0 0.1 10.0 10.0 Equations 2 a nd 3 clearly indicate that the size of the river (expressed as discharge or width) and the river slope provide important statistical M *h = 0.607 to0-823 or01 (7) explanations of migration rate. It is also evident that in this form of 29.3 11.5 17.8 analysis, the variance contribution of bank material and height are negligi- 10.0 2.5 10.0 ble, although they are likely to prove statistically significant in a group of 5.0 10.0 rivers larger than '18. In multiple regression, variables that explain relatively low proportions of the total explained variance require large data Equation 5 provides the best predictive capacity of the above set, sets in order to be identified as statistically significant. The fact that width although the unexplained variance remains high at -30%. Equation 8 is an seems to be a better predictor than is discharge probably indicates that the attempt to scale the migration rate (through channel width) to the size of use of width integrates the effect of the long-term flow record whereas the river. relatively short-term discharge records are used to calculate Q$ Q. This M h ûj 0.605 r)-0.177 must be of some comfort to those wishing to estimate bend migration rates *' = 0.018 50 (8) on ungauged rivers. W The product of discharge and slope in equation 2 is, essentially, 36.5 8.6 27.9 stream power, and, if included as such in the regression analysis, it explains 1.0 1.0 1.0 48% of the migration rate variance whereas grain size and bank height 2.0 5.0 remain nonsignificant. Stream power per unit area is a poor predictor of channel migration rate, explaining only 20% of the variance, and grain size Here, the size of the basal sediments is shown to be important in and bank height diop out of the analysis as nonsignificant (equation 4). explaining the volumetric erosion rates of river bends. In other words,

Figure 5. A schematic diagram of a channel cross section through the erosional axis (Hickin, 1974) of a meander bend which shows the large extent of boundary shear operating on the outer bed (B) compared to the outer bank (A). Note that the root zone includes only the upper part of the near-vertical cutbank and does not effect the bed of the channel.

holding river scale constant results in variations in the size of basal sedi- DISCUSSION AND CONCLUSIONS ments that have a relatively important effect on migration rate, an observa- tion also made for British rivers by Hooke (1980). These results demonstrate that mean bank erosion and mean channel In another paper, we have demonstrated a rationally derived relation- migration are predictable from channel flow and geomorphological and ship between grain size and the derived shear strength of the sediments sedimentological data. Discharge, power, or even channel width explain (Hickin and Nanson, 1984). Sediment ranged in size from clay to boulders >45% of the variance in volumetric erosion of the outer bank. This may be in a complex erosion relationship similar to that of the Hjulstrom sediment related to the physics of flow, such as the development of large-scale entrainment curve (Hjulstrom, 1935; Shields, 1936) and, therefore, was turbulence in the form of helicoidal flow, but of over-riding importance not amenable to description using linear regression. In the present paper, must be the total erosive energy available to large rivers. In combination however, the 18 river reaches were purposefully selected to lie within the with measurements of sediment size at the base of the outer bank, river fine-sand to cobble range and thereby to exhibit an essentially linear scale explains almost 70% of the total variance. Indeed, if estimates of the relationship between £>50 of the basal sediments and river bank resistance independent variables of discharge, power, and sediment size could be to erosion. As shown in equations 2, 3, 5, and 6, river size is the most determined more precisely than was possible for this study, then the level important contributor to channel migration. If volumetric erosion rate of explanation would be even higher. (M *h) were to be scaled by stream power, however, it could provide a Holding river scale constant, equations 8 and 9 show the size of basal basis for examining the relatively small influence of basal sediments on the sediment in the outer bank to be particularly influential in determining rate of lateral erosion, using the equation erosion rate. The work of Smith (1976) and Zimmerman and others (1967) suggests, however, that vegetation is also important in controlling

KA-VD a bank erosion. Variations in floodplain and bank vegetation may explain n ~ ^50- some of the scatter in Figure 4, but we did not have sufficient data to test Furthermore, if the dependent variable were to be inverted, it would this possibility statistically. For the following reasons, it does not appear, become an expression of bank resistance (equation 9), a parameter which however, that vegetation is of great importance in limiting bank erosion on in fact, has the dimensions of shear stress (force/area; Nm~2) (Hickin and these relatively large, perennial, sand and gravel rivers. Newly exposed Nanson, 1984). sand and gravel is not a suitable substrate for rapid and effective establish- n _n ment of vegetation. Even if present, vegetation protects only the subaerial = 185.78 ¿>50 where Tb (9) portion of a bank, leaving the subaqueous portion fully exposed to bound- M*h ~M*h 61.1 61.1 ary shear. Furthermore, on forested flood plains (as most of these are), the 1.0 1.0 root zone does not extend more than 1 to 2 m in depth, leaving the lower 0.1 strata unprotected by this form of rip-rap. Murgatroyd and Ternan (1983) have shown that afforestation of flood plains leads to greater bank erosion Equation 9 shows a strong positive correlation between the size of as a result of the suppression of thick gTass turf on the banks. Smith's basal sediments and bank resistance (T(,) for basal sediments coarser than (1976) experimental work on the role of roots in retarding bank erosion silt (Fig. 4). Although rj, has the dimensions of force/area, it is a coeffi- did not take into account the unvegetated alluvium below both the root cient of resistance to lateral migration, presumably dependent largely on zone and the waterline. bank strength but also absorbing all the other factors, including the statisti- Numerous cross sections surveyed in meander bends show that lateral cal variability in stream power and migration rates. In this regard it is channel migration involves substantial erosion of the laterally sloping bed analogous to Manning's n, a coefficient of total flow resistance, but not in the outer part of the channel as well as erosion of the nearly vertical directly measurable. upper part of the bank (Fig. 5) (also see Thorne and Lewin, 1979; Thome

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and Tovey, 1981). The use of a substantial vertical exaggeration in con- s seconds structing most channel cross sections has over-emphasized the predomi- nance of the banks within these sections. Indeed, during lateral migration, S water-surface slope bed erosion may be required to move up to half the vertical extent of flood-plain sediment. Once the outer bank is undermined by this process, W Watts even well vegetated and cohesive upper sediments will collapse into the river and disintegrate. As shown schematically in Figure 5, bank erosion W channel width and channel migration is probably largely determined by bed-material transport. It is for this reason that a simple relationship involving stream yr year power and basal sediment size provides such an effective means of express- ing the driving and resisting forces in this predictive model of channel p water density migration. Indeed, recent work by Neil (1984) has shown that if flood-plain alluvium is differentiated into that derived from bedload versus Tb bank strength (Nm~2) that from suspended load, then bedload transport rates can be accurately predicted from measurements of lateral channel migration. The model n total stream power per unit channel length (PFnr1) presented here projjerly addresses a bedload transport process (Bagnold, 1980) rather than one necessarily involving bank shear as assumed by co stream power per unit bed area ( fVm 2) Begin (1981).

REFERENCES CITED

ACKNOWLEDGMENTS Bagnold, R. A., 1980, An empirical correlation of bedload transport rates in flumes and natural rivers: Proceedings of Royal Society London A372, p. 453-473. Begin, Z. B., 1981, Stream curvature and bank erosion: A model based on the momentum equation: Journal of< leology, We thankfully acknowledge the support of Ron Smith and the De- v. 89, p. 497-504. Bray, D. I., 1975, Representative discharges for gravel-bed rivers in Alberta, Canada: Journal of Hydrology, v. 27, partment of Geography at the University of Alberta for the use of that p. 143-153. University's airphoto collection and to Ethel Lee of the University of 1979, Estimating average velocity in gravel-bed rivers: American Society of Civil Engineers Proceedings, Journal Hydraulics Division, v. 105, p. 1103-1122. Wollongong for assistance with computing. We are particularly grateful to Brice, J. C., 1973, Meandering pattern of the White River in Indiana—An analysis, in Morisawa, M., ed. Fluvial geomorphology: Binghamton, New York, Publications in Geomorphology, p. 176-200. John Bridge of the: State University of New York, Binghamton, for his Carey, W. C., 1969, Formation of floodplain lands: American Society of Civil Engineers Proceedings, Journal Hydraulics comments on drafts of the manuscript. The Department of Geography at Division, v. 95, p. 981-994. Daniel, J. F., 1971, Channel movement of meandering Indiana streams: U.S. Geological Survey Professionil Paper Simon Fraser University provided facilities to the first author while on 732-A, p. A1-A18. Hickin, E. J., 1974, The development of meanders in natural river-channels: American Journal of Science, v. 274, study leave, and the study was funded by a grant (no. A8376) to the p. 414-442. second author from the National Science and Engineering Research 1977, Hydraulic factors controlling channel migration, in Proceedings, Guelph Symposium in Geomorphology, 5th, Norwich, England: Geomorphology Abstracts, p. 59-66. Council of Canada. Hickin, E. J., and Nanson, G. C., 1975, The character of channel migration on the Beatton River, northeut British Columbia, Canada: Geological Society of America Bulletin, v. 86, p. 487-494. 1984, Lateral migration of river bends: American Society of Civil Engineers Proceedings, Journal Hydraulic Engineering, v. 110, p. 1557-1567. NOTATIONS Hjulstrom, F., 1935, Studies of the morphological activity of rivers as illustrated by the River Fyris: Bulletin G xjlogical Institute, University Uppsala, v. 25, p. 221-527. Hooke, J. M., 1979, An analysis of the processes of river bank erosion: Journal of Hydrology, v. 42, p. 39-62. SI units are used throughout except where otherwise stated. Symbols 1980, Magnitude and distribution of rates of river bank erosion: Earth Surface Processes, v. 5, p. 143-157. Hooke, R. LeB., 1975, Distribution of sediment transport and shear stress in a meander bend: Journal of Geolojy, v. 83, that have more than one meaning will be clear in context. A bar over the p. 543-565. Knighton, A. D., 1973, Riverbank erosion in relation to streamflow conditions, River Bollin-Dean, Cheshire: East symbol denotes the arithmetic mean. Midlands Geographer, v. 5, p. 416-426. Leighly, J., 1936, Meandering arroyos of the dry southwest: Geographical Review, v. 26, p. 270-282. Murgatroyd, A. L, and Teroan, J. L, 1983, The impact of afforestation on stream bank erosion and channel form: Earth Z)5o the 50th percentile of a grain diameter distribution (mm) Surface Processes and Landforms, v. 8, p. 357-369. Nanson, G. C., 1980, A regional trend to meander migration: Journal of Geology, v. 88, p. 100-108. Nanson, G. C., and Hicldn, E. J., 1983, Channel migration and incision on the Beatton River. American Society of Civil g gravitational acceleration Engineers Proceedings, Journal Hydraulic Engineering, v. 109, p. 327-337. Neil, C. R., 1984, Bank erosion vs. bedload transport in a gravel river: Rivers 83, in Proceedings, American Society of Civil Engineers and International Association of Hydraulic Research Conference, New Orleans: p. 204-:!ll. Outhet, D. N., 1974, Progress report on bank erosion studies in the Mackenzie River ddta, N.W.T.: Hydrological Aspects h height of the outer bank (m) ofNorthem Pipeline Development, Task Force on Northern Oil Development Report 74-12, Information Canada, p. 303-345. Page, K., and Nanson, G. C., 1982, Concave bank benches and associated floodplain formation: Earth Surface 1 Processes m metres and Landforms, v. 7, p. 529-542. Shields, A., 1936, Anwendung der Aehnlichkeitsmechanik und der turbulenzforschung auf die geschiebebcwegung: Mitteilung der Preussischen versuchsanstalt fuer Wasserbau und Schifibau, Heft 26, Berlin. Smith, D. G., 1976, EfFect of vegetation on lateral migration of anastomosed channels of a glacial meltwaler river M lateral migration rate of the channel along the erosional axis of Geological Society of America Bulletin, v. 85, p. 857-860. the bend (m yr-1) Thome, C. R., and Lewin, J., 1979, Bank processes, bed material movement and planform development in a meandering river, in Rhodes, D. D., and Williams, G. P., eds., Adjustments of the fluvial system: Dubuque, Iowa, (eiidall/ Hunt, p. 117-137. M* as above but for bends where 2.0

Qs.o 5-yr flood discharge (m3s_1) MANUSCRTFT RECEIVED BY THE SOCIETY DECEMBER 7,1983 REVISED MANUSCRIPT RECEIVED JUNE 17,1985 r radius of channel curvature (m) MANUSCRIPT ACCEPTED NOVEMBER 15,1985 Printed ill U.S.A.

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