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CATENAELSEVIER Catena 26 (1996) 85-98

Multi-channel patterns of bedrock rivers: Anexample from the central Narmada basin, India

Vishwas S. Kale a, Victor R. Baker b, Sheila Mishra Department of Geography, University of Pune, Pune 411007, India

b Department of Geosciences, Universiry of Arizona, Tucso n, AZ g5721, USA Department of Archaeology, Deccan College, Pune 411006. IndiaReceived 25 January 1995; accepted 21 August 1995

AbstractAn anomalous multiple channel pattern in bedrock is observed on a predominantly downcut-

ting reach of the Narmada River. The multi-channel reach (800-2750 m in width and 8500 m inlength) is bounded by major faul ts, and is underlain by granite and gneiss bedrock. Geomorpho-logical investigations reveal differences among the upstream, middle and downstream sub-reachesof the multi-channel study area. Whereas the upstream sub-reach is dominated by deep flows andfine sediments, the lower sub-reach is characterized by a steep gradient and rapids. The middlesub-reach is the widest, and is marked by thickly forested islands and boulder berms. Thecharacteris tics of the three different sub-reaches suggest control by the interactions of lithology,flood processes and tectonics. Estimations of Hacks (1973) stream-gradient index values indicateconsiderable variations for the SL values along the length of Narmada River. The highest value ofgradient index (SL = 797) is associated with the multi-channel reach, implying lithologic ortectonic control. Given the dimensions of the reach and its channels, i t appears that the presenthydrological regime is inadequate to produce the feature. We hypothesize that the multi-channelpattern development in bedrock was initiated by block or domal upl ift. Enhanced gradients andextreme floods permitted the system to exploit linear weaknesses in the bedrock, leading to thedevelopment of anabranches and establishment of multiple channels in bedrock. Abrupt changes inthe channel planform and morphology at the study site indicate that the river is adjusting itschannel geometry (width, depth, gradient and plan configuration) to a new equilibrium channelmorphology through the action of the extreme floods characteristic of this fluvial environment.

1. Introduction

River channel planforms in alluvial and bedrock channels have generally beenconsidered to be fundamentally different in character (Howard, 1980; Schumm, 1985;0341-8162/96/$15.00 Q 1996 Elsevier Scienc e B.V. All rights reservedSSDl 034 l-8 I62(95)00035-6



86 V.S. Kale et al./ Catena 26 (1996) M-98

Ashley et al., 1988). However, the great preponderance of channel morphologicalstudies have been conducted in alluvial channels. Single and multip le channel systemshave been identified to characterize the river morphology and channel geometry ofalluvial rivers (Schumm, 1985). In alluv ial channels, two possible mu ltiple channelpatterns are usually recognized: braided and anastomosed (Leopold and Wolman, 1957;Fairbridge, 1968). In recent years, a clear distinction has been made between the two(Rust, 1978; Mill er, 1991a). Ouchi (1985) used the term reticula te pattern to represent awidespread mul tiple channel network that has an angular cross-channel development.Recently Nanson and Knighton (1996) proposed a general classification of anabranchingalluvia l patterns, defined as systems of mul tiple channels separated by islands. Severalother varieties of alluvial anabranching rivers are recognized as well.

The term anastomosis has been used to designate mult iple interconnected channel-ways whether in alluvium or bedrock. For example, Garner (1974) follows Bretz (1923)defining an anastomosing channel system as ... an erosionally developed network ofchannels in which the insular flow obstructions represent relict topographic highs andoften consist of bedrock. In the Channeled Scabland Bretz 1923, Bretz, 1928 recog-nized anastomosing patterns as the result of cataclysmic flood water invad ing re lativelysmall preflood valleys. Spi lling of water across divides generates the anastomosingpattern as a kind of channel overfitness (Baker, 1978) using the terminology of Dury(I 964).

Relatively few studies have considered the development of river channel planforms inbedrock. Among these are works by Kirkby (1972), Shepherd and Schumm (1974)Braun (1983), Kelsey (1988), Ashley et al. (1988), Harden (1990), Seid l and Dietrich(1992) Wohl, 1992, 1993, Wohl et al. (1994) and Lorenc et al. (1994). The emphasishas been on single channels and particularly the development of meanders. As in alluvialrivers, channel pattern and associated hydraulic geometry in bedrock probably representan equilibrium between hydrologic regime and the geologic environment (Begin andSchumm, 1984), reflecting the rivers adjustment in energy expenditure (Chang, 1979).Studies of the development of meanders and multip le channel patterns in bedrock aretherefore, necessary for understanding their control by bedrock characteristics as well asby the hydraulic regime. The major objectives of this paper are to report and explain themorphologic character of the multi-channel reach of the Narmada River in central India.

2. Geological and geomorphological settingThe multi-channel reach is located at the upstream end of the incised and structurally

controlled section of the Punasa Gorge, in the central Narmada Basin (Fig. 1).The Narmada River occupies the Son-Narmada-Tapi (SONATA) lineament zone, a

megatectonic feature (Kale, 1989; Brahman, 1990; Ravi Shankar, 1991). The zoneconsists of several longi tudinal fault-bound blocks that have an episodic history ofvertical and lateral movements. Strong neotectonism and moderate seismicity character-ize the SONATA zone (Ravi Shankar, 1991). Fig. 2 provides an outline geologica l mapof the central Narmada Basin. Three major faults and a number of cross faults traversethe area. The Narmada River and its main southern tributaries, the Ajnal and the



VS. Kale ef al. / Cutena 26 (1996) 85-98 87

Fig. 1 . Location map.

Machak, show strong alignments along the regional geofracture system. The alternationof structurally contro lled bedrock gorges with alluvial reaches probably reflects differen-tia l movement between blocks bounded by major faults.

3. Reach characteristics

Channel banks upstream and immediately downstream of the study reach aretypically composed of thick Pleistocene a lluv ium. Just upstream of Joga Kalan (Fig. l ),the river, which is aligned along a major fault , turns abruptly southward and, aftercutting through a ridge of chert breccia, widens and splits into mul tiple channels overbasement granitic gneiss and gneissic granite. Further downstream chert breccia isexposed in the channel.

The maximum width of the study reach is about 2750 m, and the length is about 8500m. Further downstream a knick poin t is located at the head of a 20-m deep bedrockgorge. The average gradient of this reach is 0.0012, and at rapids it exceeds 0.03. Themean sinuosity of the study reach does not exceed 1.2. However, upstream of Handia,the alluv ial channel of Narmada is highly sinuous (P = 1.3 to nearly 1.5). The gaugingstation located at Handia, about 25 km upstream of the study reach, recorded a peakdischarge of 46,000 m3 s- on August 19, 1984. Local inquiries reveal that while thecross-channels located on the islands are active during monsoon floods, the interchannel



88 VS. Kde et al. / Catenn 26 (1996) 85-98I

! m 6L3775

DECCAN TR APS [flOWs & dykes)KATKU T SANDSTONE FO RMATION

i

I MANDHATA GROUPXv m Chertbrecaa K[SHANGAD GROUP

oic Mobile BeltsEi

Oolomite )Jindhyan Supergroup EARKESAR GROUPGwalior h Bijawar Groups 0

fl Bundelkhand Gneissic Complex

Fig. 2. Geological map of central Narrnada River B asin (after Vivek Kale, 1989).

islands are tota lly submerged only during very large floods. Such large floods mostrecently occurred in 1961, 1968, 1970 and 1984.

A geomorphological map of the reach was prepared from aerial photographs(1 : 60,000), topographical maps (1 : 50,000) and Indian Remote Sensing Satellite im-agery (1 : 50,000) (Fig. 3). Observations of the morphology, sedimentology, and distribu-tion of primary and auxiliary channels were made during geomorphoiogical investiga-tions in the field. These studies reveal differences among the upstream, middle anddownstream sub-reaches of the multi-channel study area (Table 1). Whereas theupstream sub-reach is dominated by fine sediments and deep flows, the lower sub-reachis characterized by a steep gradient and rapids. The middle sub-reach is the widest, andis marked by thickly forested islands or boulder berms (Fig. 4). The islands arecomposed of flood sediments or boulders with bedrock cores. Primary channels and,cross-channels linking the primary ones, appear to be controlled by the regionallineaments at the large scale, and joints and intrusive dikes at the small scale.

4. Formation of multi-channel patterns in bedrockGiven the deeply inc ised character of the Narmada River along most of its course

(Rajaguru et al., 1995), the multi-channel reach described here is an anomaly. Given the



V.S. Kale et ol./Catena 26 (1996) 85-98 89

Fig. 3. Indian Remote Sensing satellite image (FCC) of the multi-channel study reach (February 4, 199 I I.

dimensionsof the reach and its channels, it might seem that the present hydrologregime is inadequate to produce the feature. Similar reasoning ed Garner (1966, 15to hypothesize that the anastomosingbedrock channels of the Rio Caron, Venezcwere the result of wet, tropical rivers invading landscapes reviously formed under

jcal)67)lela,arid



90 VS. Kale ef al. / Cafena 26 (1996185-98

Table 1Geomorphological characteristics of sub-reachesSub-reach Number of Maximum Features

primary widthchannels hd

Upvr 5 2350 Deep channel; quiet flows; a high proportion of fine sedimen tsMiddle 4 2750 Large vegetated inter-channel islands ; boulder berms;

incised primary and auxiliary channe lsLower 4 1900 Numerous rapids; high gradients; sandy bars with megaripples;

abraded rock surfaces and boulder berms

climatic morphogenesis. Garner (1967) describes such patterns as rivers in themaking. However, for the Narmada multi -channel patterns are not observed alongother reaches of the river; some sort of loca l control seems responsible.

The anomalous development of multip le channels in bedrock might be attributed toone or more of the three processes: (1) different ial resistance to erosion, (2) floodprocesses, and (3) tectonic movements.

4.1. Lithological and structural controlsThe multi -channel reach of the Narmada River is bounded by two major faults (Fig.

2). The lithological constituents include gneissic granite and granitic gneiss that belongto Barkesar Group (Kale, 1989). These basement rocks are characterised by joints andlineaments, and are intruded by numerous dikes (10 cm to > 8 m in width).

The abrupt change in channel pattern with an equally abrupt change in the rock typeat the study site might suggest some sort of lithologic control. However, we do not havedetai led data on rock mass strength with which to test this hypothesis.

Erosion of bedrock channels requires both weathering and detachment. Field observa-tions at the study site indicate that the basement rock is characterized by joints on whichplucking erosion can operate. Fig. 4 clearly reveals that the alignments of the primaryand cross channels are, to some extent, controlled by major lineaments. It appears thatthe system exploited and tapped linear weakness zones in the under lying granite andgneissic rocks lead ing to the establishment of mul tip le channels in bedrock. Geomorpho-logical studies by Kale and Shingade (1987) indicate that multip le bedrock channels canform by coalescence of grooves and potholes along joints in basalt bedrock. Thedevelopment of inner channels, beginning with long itudina l lineations and grooves isalso indicated by Shepherd and Schumms (1974) flume experiments. Our observationsin the middle Krishna Basin of India have revealed an association between multi-threadand wide channels, and grani tic rocks characterized by joints.Thus, litho-structural control may be important for the reach as a whole, but thisfactor alone does not explain the geomorphological and sedimentological differencesbetween the three sub-reaches.



V.S. Kale e t al./ Catena 26 (19961 85-98 91

7 7 l BHARTAR

. BHENSWARA

GEOMORPHIC MAPSCABLAND DENSE FOREST GRASSES/FINES

VilV MAJOR FA ULTS - - - LINEAMENTSFig. 4. Sketch map of the multi-channel reach showu: major geomorpho logical features.



92 VS. Kale et al. / Catena 26 (1996) 85-98

4.2. Flood processesThe Narmada River experiences occasional high-magnitude floods. During such large

floods the incised channel is overtopped at several places. Several high magnitude floodsin this century (1961, 1968, 1970, 1973 and 1984) have filled the incised channel of theNarmada River. A catastrophic flood event in such a flood-prone river is likely to giverise to spectacular erosional and depositional features, including scabland, potholes,boulder berms and inner channels (Baker, 1978; Baker and Pickup, 1987; Wohl, 1992,1993; Lorenc et al., 1994).

Evidence of high magnitude floods is present throughout the anomalous reach of theNarmada (Kale et al., 1994). Boulder berms ranging in length from 5 to > 15 m arecommonly observed in the interchannel areas. Flood debris and flood scars on treesoccur up to 13 m above the low water leve l. Simi larly , scabland-like features, f lutemarks, polished and abraded rock surfaces, and occurrence of megaripples formed ofsand (spacing = 1.5 to 3 m), suggest the effect of large floods.Although gradient is not directly determined by hydraulic regime in bedrock chan-nels, the rate of erosion depends upon flow of water and sediment over the bed(Howard, 1980). Abrasional surfaces and flute marks demonstrate that erosion ofbedrock is taking place under the present hydraulic regime. This is also supported by thepresence of large boulders of loca l orig in. The occurrence of polished and abraded rocksurfaces and potholes suggest high velocities and Froude numbers close to 1 for asustained length of time (Kale et al., 1994).

Given the high flood velocities and Froude numbers a t this reach, one would expectconsiderable erosion and sediment transportation capabilities. F lood power of the riverin these floods can be estimated from relevant parameters of the study reach, includingthe mean gradient of 0.0012. Local slopes at the rapids reach at least 0.03, and it is thesevalues that are most important for bedrock erosion and ini tiating boulder transport. Thedischarge of 46,000 m3 s- (p ea on record at Handia) and the varying channel width(800 to 2750 m) can be used to yield a rough estimate of the stream power per unit areaw according to the formula (Baker and Costa, 1987):

w= yQS/wwhere y is the specific weight of water (9800 N m-), Q is discharge (m3 s-l), S isslope and w is the water-surface width. The estimated values range between 200watts m -* for the widest sub-reach (slope = 0.0012 and width = 2750 m) and > 7000watts m - for the lower sub-reach (for slope = 0.03 and width = 1900 m). Simi larly , atthe rapids the shear stress (slope-depth product) can exceed 3000 N m-. These valuesindicate the high competence of the river during extreme floods, particularly in the lowersub-reach. The values are comparable to those noted on other rivers display ing spectacu-lar bedrock erosion (Baker and Pickup, 1987; Rathburn, 1993; OConnor, 1993).

Baker and Pickup (1987) have described the occurrence of a series of anastomosingchannels and scour pools in upper Katherine Gorge, Australia. Because of morphologicsimilarity to flood erosional forms in the Channel Scabland of Washington, they haveattributed these features to extreme flood flows. Longitudina l grooves and shallow lineardepressions para llel to flows, in Piccaninny Creek in northwestern Australia were



V.S. Kale et al. / Catena 26 (I 996) 85-98 93

inferred by Wohl (1993) to result from longi tud inal vortices and turbulent vorticesduring high-magnitude flood flows. While cataclysmic flood erosion seems clearly to beoccurring on the lower sub-reach, the upper sub-reach remains problemat ic because ofits association with fine sediments and a low proportion of rocky channels and scablandareas. Thus, flood processes alone do not seem sufficient to explain the multi -channelpattern.4.3. Tectonic processes

Fluvial anomalies, such as loca l changes in channel patterns and loca l widening ornarrowing of channels are possible indicators of active tectonics (Gregory and Schumm,1987). Channel patterns, which constitute a fourth degree of freedom in variousequilibrium schemes of river behaviour (Chang, 1979) are sensitive indicators ofvalley-slope changes (Schumm, 1986).

The gradient of bedrock channels is a semi-independent variable and is not directlydetermined by the hydraul ic regime (Howard, 1980). Factors includ ing the physicalcharacteristics of rocks and tectonic movements determine the channel gradient of thebedrock channels. In alluvial channels, with increasing shear stress (slope-depth prod-uct) high sinuosity streams transform to braided streams (Begin and Schumm, 1984).Therefore, even minor changes in the gradient of large rivers, like Narmada, are likely toupset the balance between process and form, and both the extrinsic and intrinsicgeomorphic thresholds (Schumm, 1987) will be exceeded.

Hack (1973) defined a stream-gradient index (SL) which is the product of slope of areach times the length from the headwaters divide. Variations in gradient index reflectthe longitudinal variations in discharge, but more commonly the litholog ic or tectoniccontrols on channel slope of a given reach (Hack, 1973; Bul l and Knuepfer, 1987). Fig.5 shows that the SL indices for the Narmada River range between 71 and 797. The highvalues of the index (SL = 797) for the reach between Handia and Punasa suggest thatthis may be the most actively downcutting part of the Narmada River, followed byanother reach downstream of Rajghat (SL = 774) implying tectonic or lithologiccontrol. The abrupt changes in the gradien t and SL values at the study site can be relatedto both changing bed resistance (Miller, 1991 b) and tectonic uplift (Wohl et al., 1994).

Large rivers, owing to their relative ly low channel gradients, are most significantlyaffected by the minor changes in slope induced by active deformation (Schumm, 1986).The Narmada River, with a very large discharge and an average gradient o f 0.00072 islikely to be significantly affected by any tectonically induced increase in the channelgradient. Theoretical calculations based on average width of the undivided reach (735m) and peak discharge at Handia (46,000 m3 s- > indicate that, at very low slopes, evenvery small amounts of tectonic upl ift in the river bed leve l will so increase the slope asto result in major changes in the power per uni t area of bed. A local rise in bed leve l byonly 20 cm, for example, could raise the power per unit area by a factor of 30. At thepresent site, this effect will be, obviously, downstream of the axis of upli ft.

Experimental studies and fie ld observations confirm that a change of valley-slope willcause a change of channel pattern and dimensions (Schumm, 1986). Ouchi (1985), onthe basis of exper imental studies, inferred that domal uplift across a meandering river


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94 VS. Kaie et al. / Carenn 26 11996185-981200 I-

NARMADA RIVER1000mE 800Lrz- 600k!3l-t=g 400

P = PunasaI -

/ 159. . , . . . .. . . . . . . . . ., - . . . . . . . . . . .. . . . . . . . . .............................................................

/ - .................................................. 91..................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . /I -, I0 200 400 600 000 1000 1200 1400

200

0DISTANC FROM SOURCE IN KM

Fig. 5. Profile of the Narmada River showing values of Hacks (1973) stream-gradient index (SL) ingradient-meters.

leads to the development of reticulate or anastomosed channel patterns. According tohim, on the upstream side of the upli ft, as a result of damming effects, there will beinundation of flood-plain and channel avulsions, leading to the development of reticulateor, in some cases, anastomosing channel pattern. Anastomosing channel patterns ob-served along downcutting streams in south-central Indiana were ascribed to avulsion byMil ler (1991a). Burnett and Schumm (1983), Schumm (1986) and Gregory and Schumm(1987) described examples illustrating the development of a braided pattern downstreamof the rise and anastomosing pattern upstream of the rise, as a resuit of upli ft andconsequent reduction of valley-slope above the axis of uplift. Braided channels are theresult of high bed-load transport on steep gradients with high width-to-depth ratios anderodible banks (Schumm, 1986; Miller, 1991b). At the present site, although thegradients are steeper than upstream, the banks are more resistant with the channel cutinto bedrock. Further, estimates based on hydraulic modeling suggest that the annualpeak floods on Narmada are competent to move the bed sediments up to 2.5 m inintermediate axis (Kale et al., 1994). Such a condition would not favour the developmentof a braided channel pattern for the bed sediments are entire ly reworked.

The raising of channel bed increases the specific energy (sum of pressure andvelocity heads). At the present site, the raising of the channel bed will increase thespecific energy downstream but will decrease upstream. Channel widening in conjunc-tion with incis ion, concentrated at locations of maximum boundary shear stress andstream power per unit area, is the most effective mechanism of energy dissipation in acatastrophically disturbed system (Simon, 1992). In other words, channel widening



V.S. Kale et al. / Catena 26 (1996) 85-98 95

offsets increase in the hydraulic depth caused by incision, and allows an initial increasein specific energy due to an increase n gradient to decreasewith time (Simon, 1992).It, therefore appears hat under the given water and sedimentdischargeconditions aswell as channel perimeter lithology, a catastrophically disturbed bedrock channel willestablish a width, depth, gradient and plan configuration that results in a minimumstream power (Chang, 1979; Simon, 1992) or stream power per unit length (Yang andSong, 1979).

Therefore, the formation of multiple channels in bedrock and associated localincrease n channel width and channel capacity for the Narmada River might representan important means of reducing specific energy, under high-energy conditions inducedby block or domal uplift. The associationwith the major faults and the fact that strongneotectonism and moderate seismicity characterise the Son-Narmada-Tapi (SONATA)lineament zone (Ravi Shankar, 1991) strongly imply tectonic control on the SL values aswell as on the formation of multi-channel pattern at the study site.

5. A hypothesis for the development of multi-channel bedrock patternsThe high SL index values and the occurrence of major faults up and downstreamof

the study reach, the local setting of the anomalous each and the tectonic history of thearea all suggest that the multi-channel pattern development was initiated by block ordomal uplift. Neotectonic movement of the fault-bound block would have raised the bedlevel of the Narrnada River, leading to a damming response n the upstreamsub-reach(Fig. 6). This is indicated by the occurrence of deep water, quiet flows and predomi-nanceof fine-grained sediment n the upper sub-reach.The tectonic uplift of the channelbed increases he channel gradient as well as the stream power, greatly increasing theenergy conditions downstream of the uplift. It appears hat the channel responded o thiscatastrophic disturbance by channel widening and incision, allowing specific energy todecreasewith time. This is an interesting contrast to anabranches n alluvial channels

/ \. - FAULT I

UPLIFTED BLOCKI

F,NEs LHIICLc==1 FLOW J I -. . . . . . . . . .

+/\ granite/gneisses , , i LLn,

IFAULT II@= PUNGHAT @ = BHENSWARA

BRECCIA

Fig. 6. Sketch of relationships between Narmada River profile and structural/tectonic features of the studyreach.



96 V.S. Kale et al. / Cutenu 26 (1996) 85-98

which appear to locally increase unit stream power and enhance sediment through put(Nanson and Knighton, 1996). In the downstream sub-reach of the study site, increasedgradients would have resulted in slope adjustment by degradation. This is suggested bysteeper channel slopes and numerous rapids in the lower sub-reach (Fig. 6). The knickpoint and a deep gorge located downstream of the multi-channel reach, provide evidenceof intense high-gradient conditions. With a rise in the bed level, overbank flows duringextreme floods, devo id of significant bedload and having sufficient energy to erode,must have facilitated rapid incision (particularly below the uplift) along deeply weath-ered joints and weak zones (lineaments) in the under lying granite and gneissic rocks.This led to the development of anabranches and establishment of the multip le channelsin bedrock. Thus, a reduction of specific energy was achieved by channel widening andby degradation in bedrock. The lithology of the granite and its fracture system favouredthe development, under the influence of occasional intense floods, of the multi-channelpattern. Lack of disruption for the Narmada drainage, diverting it to new locations,indicates a relatively slow uplift of the fault-bounded block.6. Conclusion

A multi-channel pattern occurring on the regionally downcutting Narmada Riverindicates probable local effects of tectonic deformation and domal up lift, as well as theinfluence of structure and flood processes. The Narmada River, which throughout itscourse is deeply incised in bedrock or alluvium, deviates from its general pattern andsuddenly widens its channel at the study site, giving rise to a multi-thread pattern inrock. Large rivers, like the Narmada, owing to their relatively low channel gradients, arelikely to be significantly affected by any tectonically induced increase in the channelgradient. The raising of the channel bed increases the specific energy downstream of theupli ft. Using the concept of minimization of stream power, Simon (1992) argues thatdegradation accompanied by widening is the most efficient means of fluvial energydissipation. Anabranching usually concentrates stream power (Nanson and Knighton,1996) and braid ing disperses it across a wider channel width. The Narmada appears tobe minimizing the stream power by adjusting its channel geometry (width, depth,gradient and plan configuration). It is generally accepted that downcutting streams aredisequilibrium systems (Bull and Knuepfer, 1987; Miller, 1991b) and, hence, a distinc-tive channel geometry determined by the local geologic and tectonic controls impliesthat the channel is attempting to attain an equilibrium channel-bed form. The develop-ment of the multi-channel pattern, caused by a variable external to the system (tectonism),provides an excel lent example of a river near its pattern threshold (Schumm, 1987). Theresponse is broadly analogous to that occurring in alluvial rivers, but the conditions offlood energy and channel boundary properties (lithology) are more extreme, resulting indifferent timescale for the equilibr ium adjustment.

AcknowledgementsThis paper is contribution No. 34 of the Arizona Laboratory for Palaeohydrological

and Hydroclimato logical Analysis (ALPHA), University of Arizona. The research was



VS. Kale et al. / Catena 26 (1996) 85-98 97

supported by Indian Department of Science and Technology Grant ESS/CA/A3-04/90to V.S.K. and S.M. The authors thank S.N. Rajaguru, Y. Enzel, L. Ely, Avij it Gupta andVivek Kale for fru itfu l discussions. Reviews by G.C. Nanson and E.E. Wohl were veryhelpful in preparing the final version of this paper.

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