Conditions for branching in depositional

Douglas J. Jerolmack* Department of Earth and Environmental Science, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA and National Center for Earth-surface Dynamics, University of Minnesota, Minneapolis, Minnesota 55414, USA David Mohrig Jackson School of Geosciences, University of Texas at Austin, Texas 78712, USA

ABSTRACT It is often taken for granted that rivers organize transport into a single active . In some net-depositional environments, however, fl ow of water and sediment is distributed in several stable channels. Such branching rivers may be confi ned in valleys (anabranching or anastomosed) or unconfi ned on deltas (), and their existence confronts us with the very basic question of what governs the spatial organization of channel patterns in sedi- mentary landscapes. Current models for equilibrium channel morphology cannot predict the occurrence of branching rivers because they do not consider dynamical processes such as , i.e., the rapid abandonment of a channel in favor of a new path at lower eleva- tion. The requisite conditions for avulsion have been the subject of ongoing debate. Here we resolve the conditions leading to channel avulsion, and show that branching rivers occur when avulsion is the dominant mechanism of lateral channel motion. A compilation of fi eld and laboratory data demonstrates that avulsion frequency scales with the time required for sedi- mentation on channel beds to produce a deposit equal to one channel depth. From the relative rates of bank and channel sedimentation, we derive a dimensionless mobility number that accurately predicts the conditions under which anabranching and channels occur. Results may be directly applied to modeling landscape evolution over human and geo- logic time scales, and for inverting formative environmental conditions from channel deposits on Earth and other planetary surfaces.

Keywords: morphodynamics, bifurcation, wandering rivers, braided, avulse, fl uvial.

INTRODUCTION PROCESS SCALING ANALYSIS plain (superelevation) equal to approximately

Anabranching rivers (or anastomosing) may We hypothesize that branching in deposi- one channel depth (Fig. 1). In this sense TA is be defi ned as a system of multiple channels tional rivers is a manifestation of avulsion, and an avulsion time scale, or more properly, it is (branches) that are separated by relatively stable, that the requisite condition for branching is that the average time between avulsions for a single often vegetated (Nanson and Knighton, the lateral motion of channels via bank erosion channel. The avulsion frequency of the

1996; Makaske, 2001; Jansen and Nanson, is small relative to their vertical displacement system, fA, may be directly estimated as: 2004). Individual branches of an anabranch- above the surrounding fl ood plain via focused vN = A ing river may be straight, sinuous, or braided sediment on the channel beds. This fA , (1) (Nanson and Knighton, 1996). Currently there idea is formalized by a simple scaling analy- h is no theory to predict under what conditions sis of the time scales associated with dominant where N is the number of active channels. We a river will form multiple branches. Neverthe- processes. We defi ne the lateral migration time defi ne the mobility number, M, of a given river less, it has been argued that anabranching rivers scale, TC = B/vC , as the time, TC, it would take a system as the ratio of avulsion and lateral- can be an equilibrium form under certain com- channel to migrate the distance equal to its total migration time scales, binations of fl ow strength, , width, B, given a representative measure of bank T h v ==A C and vegetation (Nanson and Knighton, 1996; erosion rate, vC (e.g., Tal et al., 2004). A repre- M . (2) TC B vA Jansen and Nanson, 2004). Stable distributary sentative time scale for channel , TA, channels on deltas and alluvial fans are another is similarly defi ned as the time required for the For rivers having M >> 1 we expect a system – example of branching river systems. While it is river to aggrade one average channel depth, h , with a single channel that sweeps across its known that their occurrence is associated with above the distal fl ood plain. This time scale may fl ood plain relatively rapidly, reworking the – environments of net deposition, the conditions be approximated as TA = h/vA , where vA is a fl ood plain as it moves laterally. Avulsions are that cause channels to become distributary are representative aggradation rate. Because deposi- expected to be relatively rare (Fig. 1), while not well quantifi ed. Current models for equilib- tion rates on the distal fl ood plain are often very rapid combing would tend to cause coales- rium channel morphology (Parker et al., 1998; small relative to those measured in the vicinity cence rather than creation of channels. For Dade, 2000) cannot predict the occurrence of of a channel (Heller and Paola, 1996, Törnqvist cases of M << 1 individual channels aggrade branching rivers because they do not consider and Bridge, 2002), vA is taken to equal a near- rapidly compared to any lateral shifting in dynamical processes such as avulsion, i.e., the channel or in-channel rate of sediment accu- response to bank erosion (Fig. 1). Avulsions rapid abandonment of a channel in favor of a mulation. In an examination of ancient river should be relatively frequent, causing several new path at lower elevation. deposits, Mohrig et al. (2000) found that avul- channels to be active at once (Makaske, 2001). sion was associated with aggradation of the river Our hypothesis predicts that these rivers should *E-mail: [email protected]. channel to a height above the surrounding fl ood be branched. If M is 0 (10°) (order one), transi-

© 2007 The Geological Society of America. For permission to copy, contact Copyright Permissions, GSA, or [email protected]. GEOLOGY,Geology, May May 2007; 2007 v. 35; no. 5; p. 463–466; doi: 10.1130/G23308A.1; 3 fi gures; Data Repository item 2007108. 463 ′ B M >> 1 A A′ A 104 h ] A -1 2 Channel belt [yr 10 2

A A R = 0.85 deposits f 100 B = b1 + b2 FLOW b1 b2 SE C′ C -2 M << 1 C′ 10 Field data

Calculated Lab data 10-4 C 10-4 10-2 100 102 104 -1 Measured fA [yr ] 103 Figure 1. End-member , deposits, and defi nitions. Top: Hypothetical single- Single- channel meandering river in planform (left), and representative cross section through ε Branching channel 2 belt (right). Lateral channel movement is accomplished principally by bank erosion. Bottom: 10 Anabranching river with multiple stable branches. Cross section at right shows that these B – channels are superelevated (SE), with relative superelevation SE /h = 1. River represents 101 N avulsion-driven morphology. Note that Bb= ∑ is total wetted width of all active channels, Multiple thread i i 0 where N is number of active channels and bi is width of each channel. M is mobility number. 10

10-1 Single Parker criterion, thread tion between single and multiple branches is which were computed using a variety of meth- 10-2 expected if avulsion and lateral bank erosion ods including dated cores in channel beds and 10-2 10-1 100 101 102 10 are the only important processes. banks, aerial photography, and channel surveys Mobility #, M Parker (1976) provided a theoretical stability (see footnote 1). Note several caveats against 104 criterion, ε, that can be used to assess whether overinterpretation of specifi c values reported C a river should be braided (i.e., a single channel for mobility number. Channel geometry was [m] 103 2 with multiple threads of high-velocity fl ow), measured from modern rivers, so our analysis B R = 0.92 or a single-thread channel with a sinuous to implicitly assumes that these values have not straight plan form (Fig. 2). This criterion may changed substantially over the Holocene. No be written as ε=S ghB4 Q, where S, g, and quality checks were performed on reported 102 Q are water surface slope, gravitational accel- rates, or on ages used to derive rates. Some eration, and forma tive water , respec- researchers may object to reducing the large Calculated tively. A single-thread channel is the dominant variability in any natural river to a single datum; 101 form at ε << 1, while a braided form is the com- in reality, a river system with varying channel 101 102 103 104 Measured B [m] mon channel type for ε ≥ 1 (Parker, 1976; Dade, geometry and rates of evolution could have a 2000). We propose that the mobility number spatially varying mobility number. A compre- Figure 2. Measured and computed river and the Parker stability criterion when taken hensive error analysis of the data is beyond the characteristics (see text and footnote 1). A: together provide a complete qualitative picture scope of this paper; however, it is not necessary Measured versus calculated avulsion fre- quency (1) for fi eld and laboratory data. Line of expected river planform pattern. for our purposes here. Validation of the scaling denotes perfect agreement. B: Phase space argument requires only that the predicted lim- of mobility number (2) and Parker (1976) SCALING TEST its (branching for M << 1; single channel for stability criterion. Lines delineate channel To test whether the derived mobility number is M >> 1) are consistent with the data. patterns observed from data. Diamonds and capable of discriminating between single-channel Because avulsion is an inherently stochas- circles indicate sinuous (single thread) and braided rivers, respectively; white, black, and and branching systems, we have compiled data tic process, the computed avulsion time should gray symbols are single-channel, branch- from 30 net-depositional systems reported in the be seen as an average value. In reality, avulsion ing, and transitional systems, respectively. literature (see GSA Data Repository1). These may occur at superelevations >1 or <1, and one C: Measured versus calculated total chan- data include lowland rivers, deltas, and alluvial must integrate over many avulsions to recover nel width for fi eld data only. R-square values associated with the proposed relations for fans with perennial to ephemeral discharges, the expected scaling behavior. To test the validity avulsion frequency and channel width are located in tropical to arid environments and of our avulsion frequency relation (Equation 1), shown in A and C, respectively. transporting particles ranging from fi ne sand we compare computed values to average values to cobbles. The reported systems also cover a measured in natural and laboratory systems. The wide range of scales, from the laboratory to the agreement demonstrates (Fig. 2A) that long-term . The size of the data set analyzed avulsion frequency may be estimated using the Paola, 1996; Mohrig et al., 2000; Törnqvist and here is limited by the number of studies provid- simple relation (Equation 1) proposed here. This Bridge, 2002; Ashworth et al., 2004; Slingerland ing enough information to estimate vC and vA, result is of great signifi cance because model- and Smith, 2004). Varying degrees of importance ing efforts have shown that the morphology and have also been assigned to the frequency of avul- 1GSA Data Repository item 2007108, river data depositional patterns of channelized landscapes sion triggers such as fl oods, ice jams, and animal and descriptions of how these data were derived, is are very sensitive to avulsion frequency (Heller disturbance (Slingerland and Smith, 2004). Given available online at www.geosociety.org/pubs/ft2007. htm, or on request from [email protected] or and Paola, 1996; Sun et al., 2002), and there that our data come from fi eld and laboratory set- Documents Secretary, GSA, P.O. Box 9140, Boulder, are competing descriptions of conditions neces- tings representing the broadest possible range CO 80301, USA. sary for avulsion (Bryant et al., 1995; Heller and of depositional environments, it appears that (1)

464 GEOLOGY, May 2007 knowledge of the style of avulsion trigger is not for river management and habitat issues, and necessary for modeling the long-term evolution is equally important in the reconstruction of of channel deposits, implying that occurrence of paleoenvironmental conditions via the analysis a triggering event is not a limiting factor in avul- of ancient fl uvial deposits. sion frequency; and (2) the condition for avulsion is well approximated by the time required to EQUILIBRIUM CHANNELS AND achieve a relative superelevation of one (Fig. 1). BRANCHING These results may be applied directly to morpho- Assuming steady uniform fl ow and a dynamic models of channel and fan development Chezy fl ow-resistance relationship, channel (Sun et al., 2002, Swenson, 2005). 50 km width of a single-branch river may be cast as Yellow River; M = 16 = 12 33θ 3 All reported river systems with a single active B cQSRf dg50 (Parker et al., 1998; 1 θ channel correspond to M > 10 (see footnote 1). Dade, 2000), where , d50, and cf are dimension- All systems that clearly exhibit a branching pat- less bed (Shields) stress, median grain diam- tern have M < 10°. Most river systems having eter, and Chezy friction factor, respectively, 100 ≤ M ≤ 101 are transitional, generally contain- and R = 1.65 is the relative specifi c gravity ing one dominant channel and a small number of for quartz. The measured values for total wet- secondary channels. Using measured river data ted width of rivers in our database are plotted (see footnote 1) we have constructed a phase against calculated values of single-channel space that relates to M and ε. We width in Figure 2C. These data show that the fi nd this method effectively segregates systems sum of active channels in anabranching and dis- into four categories: single channels with mul- rivers behaves like a single channel of tiple threads (simple braided channels), single comparable size. More support for this idea may channels with single threads (straight to sinu- 100 m be found in the mean values for Shields stress ous channels), multiple channels with multiple Cooper Creek; M = 0.3 of sandy and gravel-bedded branched rivers (1.1 threads (branching systems with braided chan- and 0.07, respectively), which are very close to nels), and multiple channels with single threads 2 km the constant observed values for single-channel (simple branching systems) (Fig. 2B). Because systems (Parker et al., 1998). The mean channel of the variable data quality these values for M depth, total wetted width, and dimensionless bed Niobrara River; M = 0.2 should be taken only as order of magnitude stress may all be estimated (Parker et al., 1998; estimates. Nonetheless, analysis of these data Figure 3. Example rivers and their mobility Dade, 2000) without reference to the number clearly demonstrates the discriminating power numbers (M). Yellow River image is from of channels in a river system, indicating similar of the mobility number. http://visibleearth.nasa.gov; Niobrara River channel-forming processes for all types. image is from http://www.dnr.state.ne.us/ It appears that the pattern of branching has CASE STUDIES databank/spat.html; and Cooper Creek image little to do with the mean transport of water and is courtesy of G. Nanson. It is useful to focus on a few specifi c river sediment, and therefore it is not likely an adjust- systems to address the meaning of computed ment to optimize fl ow effi ciency or transport mobility numbers (see footnote 1). The Yellow capacity (e.g., Jansen and Nanson, 2004). Our River of China has an extremely large Holo- mm/yr is used, mobility number is still <1 analysis suggests that the cause of branching is

cene channel aggradation rate of vA = 100 (M = 0.7), and it is probably much less due to independent from the causes of other river pat-

mm/yr (Fig. 3). Although this river avulsed on the (likely) overestimated value for vC. Cooper terns such as braiding. Braiding results from decadal time scales before human intervention, Creek shows that very stable channel banks deposition, a fundamental instability of constructing a large delta, only one channel may induce an avulsion-driven morphology, coupled fl uid and sediment transport within a has been active at any one time due to the high even if channel aggradation rates and avulsion channel (Parker, 1976). While bars may cause degree of channel migration by bank erosion frequency are very low. It is relative rates, not fl ow splitting at small to intermediate scales, we (Wu et al., 2005). Despite modern channel con- absolute rates that are important. A fi nal case argue that branching (and anastomosis) in depo- trol structures, the mouth of the Yellow River is the lower Niobrara River, Nebraska (Fig. 3). sitional rivers derives from an emergent instabil- exhibits very similar activity today (van Gelder The channel has undergone rapid bed aggrada- ity at a larger scale , i.e., channel avulsion. The et al., 1994). The values M = 16 and ε = 0.07 tion throughout the past 50 yr in response to existence of multiple stable branches in the riv- indicate that the Yellow should be a base-level rise from reservoir construction. ers examined here appears to be determined by single-channel, sinuous river system (Fig. 2), in This aggradation has driven multiple avul- boundary conditions conducive to net deposi- agreement with observation. Hence, this exam- sions and the conversion of some reaches to tion of sediment, combined with some degree of ple serves to demonstrate that rapid aggrada- narrower anabranches separated by stable, channel-bank stability (usually from vegetation) tion alone is not suffi cient to generate a branch- vegetated islands (Ethridge et al., 1999). Our such that the mobility number is 0 (10°) or less. ing system. An interesting counter example is calculations suggest that the present morphol- This point addresses the question as to whether the anabranching Cooper Creek (Gibling et al., ogy of the lower Niobrara should be branch- anabranching patterns can exist as a persistent 1998; Fagan and Nanson, 2004), in arid central ing and braided (M = 0.2, ε = 2), in agreement channel-system form (Makaske, 2001; Jansen (Fig. 3). Although vertical aggrada- with observations. We propose that the mobil- and Nanson, 2004). Anabranching need not be ≤ tion is very small (vA 0.1 mm/yr), the exceed- ity number may be used to predict if a change merely a transitional pattern; the requirement ingly low channel migration rate, estimated as in boundary conditions, whether tectonic, cli- that M < 1, however, means that anabranching ≤ vC 8 mm/yr, results in a maximum value for matic, eustatic, or anthropogenic, will result in may only be maintained in environments under- M of 0.3, indicating multiple branches. Even a conversion from single-channel to branch- going bed aggradation rates that are rapid com-

if the lower (fl ood plain ) estimate vA = 0.04 ing morphology. This is of critical importance pared to lateral cutting rates.

GEOLOGY, May 2007 465 CONCLUSIONS surface Dynamics under agreement EAR-0120914. Murray, A.B., and Paola, C., 2003, Modelling the Paul Heller and Torbjörn Törnqvist gave thorough and effect of vegetation on channel pattern in bed- By deriving the characteristic time scales of constructive reviews that substantially improved the load rivers: Earth Surface Processes and Land- dynamical processes, we determined the con- manuscript. We gratefully acknowledge the careful forms, v. 28, p. 131–143, doi: 10.1002/esp.428. ditions leading to the formation of branching work of previous authors that allowed the compila- Nanson, G.C., and Knighton, A.D., 1996, Ana- patterns in depositional rivers. Anastomosed tion of our database. 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