Model for the Formation of Single‐Thread Rivers in Barren Landscapes and Implications for Pre‐Silurian and Martian Fluvial Deposits

Flume experiments and field observations show that bank vegetation promotes the formation of narrow and deep single‐thread channels by strengthening riverbanks. Consistent with this idea, the pre‐Silurian fluvial record generally consists of wide monotonous sand bodies often interpreted as deposits of shallow braided rivers, whereas single‐thread rivers with muddy floodplains become more recognizable in Silurian and younger rocks. This shift in the architecture of fluvial deposits has been interpreted as reflecting the rise of single‐thread rivers enabled by plant life. The deposits of some single‐thread rivers, however, have been recognized in pre‐Silurian rocks, and recent field studies have identified meandering rivers in modern unvegetated environments. Furthermore, single‐thread‐river deposits have been identified on Mars, where macroscopic plants most likely never evolved. Here we seek to understand the formation of those rarely recognized and poorly characterized single‐thread rivers in unvegetated landscapes. Specifically, we quantitatively explore the hypothesis that cohesive muddy banks alone may enable the formation of single‐thread rivers in the absence of plants. We combine open‐channel hydraulics and a physics‐based erosion model applicable to a variety of bank sediments to predict the formation of unvegetated single‐thread rivers. Consistent with recent flume experiments and field observations, results indicate that single‐thread rivers may form readily within muddy banks. Our model has direct implications for the quantification of riverbank strengthening by vegetation, understanding the hydraulic geometry of modern and ancient unvegetated rivers, interpreting pre‐Silurian fluvial deposits, and unraveling the hydrologic and climate history of Mars.


Single-Thread and Meandering Rivers
Single-thread alluvial rivers are relatively deep and narrow, straight to sinuous streams that carry water and sediments downslope within a single channel. Among single-thread rivers, meandering rivers feature sinuous bends that grow in amplitude to the point of cutoff and maintain strikingly regular planform geometry across scales while snaking through low-lying floodplains (Leopold & Wolman, 1960;Williams, 1986). Meandering rivers migrate laterally at rates that generally increase with channel size and planform curvature (Braudrick et al., 2009;Hickin & Nanson, 1984;Howard & Knutson, 1984;Sylvester et al., 2019). Throughout this manuscript, we use "single-thread" as a sole descriptor of planform geometry; "meandering" is used for single-thread streams with bends that grow due to lateral migration.
In channel bends, flow inertia and scour lead to deeper flows along the outer bank (Bathurst et al., 1977). The resulting lateral pressure gradient leads to an in-channel helical flow that entrains sediment along the outer bank and transports it toward the inner bank further downstream, where it gets redeposited in the form of bank-attached bars (i.e., point bars; Figure 1a; Bluck, 1971;Jackson, 1975Jackson, , 1976Dietrich et al., 1979). To sustain meandering and to prevent the formation of midchannel bars, single-thread rivers require the formation of one main helical-flow cell across the width of a relatively narrow and deep channel. Thus, meandering rivers typically have low width-to-depth ratios (conservatively with W/h <~200 and often in the 20-100 range; Engelund & Skovgaard, 1973;Parker, 1976;McLelland et al., 1999;Gibling, 2006). Sinuous but nonmeandering single-thread channels also have W/h <~200, and straight single-thread rivers tend to form with even lower W/h (typically W/h < 10; Parker, 1976). The W/h limit for river meandering (and thus single threads) arises from a midchannel bar instability that is well understood theoretically (e.g., Crosato & Mosselman, 2009;Seminara & Tubino, 1989;Struiksma et al., 1985). Ultimately, because deeper flows impart greater shear stresses upon the riverbed and banks, the formation of single-thread rivers is intimately tied to the ability of riverbanks to resist erosion and confine relatively deep flows within a single relatively narrow channel (e.g., Ferguson, 1987;Schumm, 1960Schumm, , 1963. Hence, natural single-thread alluvial streams tend to form in low-relief alluvial plains where bed stresses are buffered by gentle channel gradients (Ashmore, 1991;Smith & Smith, 1984) and where formative-discharge floods do not result in bank breach, channel branching, and eventual river braiding (Abdullatif, 1989;Bluck, 1974;Takagi et al., 2007).
Previous studies suggested that both higher mud contents in riverbanks (e.g., Schumm, 1960) and dense bank-vegetation covers (Graf, 1978;Huang & Nanson, 1998;Micheli & Kirchner, 2002;Murray & Paola, 2003;Smith, 1976) may slow bank erosion and lead to lower W/h. Even a thin cover of plants with shallow roots seems to promote the formation of meandering rivers within otherwise cohesionless banks, perhaps through the armoring effect of vegetated slump blocks along the rivers' outer banks (e.g., Micheli et al., 2004;Parker et al., 2011). Untangling the effects of fine cohesive sediments and vegetation on natural rivers, however, is challenging. In addition to field studies and despite the difficulty of scaling bank erosion down to laboratory scales (e.g., Kleinhans et al., 2014;Kleinhans et al., 2015), some flume experiments have shown that either cohesive sediments (e.g., Peakall et al., 2007;van Dijk et al., 2012van Dijk et al., , 2013 or bank vegetation (e.g., Braudrick et al., 2009;Gran & Paola, 2001;Tal & Paola, 2007) can promote the formation of deeper and narrower, single-thread, and even meandering, channels. Given the ubiquity of plants in most terrestrial environments, bank vegetation likely provides a major control on the hydraulic geometry of modern rivers (e.g., Dietrich & Perron, 2006;Gurnell, 2014).
The relationship between bank vegetation and river planform geometry on modern Earth raises the question of what rivers might have looked like before the Silurian emergence of vascular plants (e.g., Cotter, 1978;de Almeida et al., 2016;Schumm, 1968). As a plant-devoid endmember, pre-Silurian river deposits may offer the opportunity to better constrain physical and biogeomorphic controls on river patterns. However, interpreting prevegetation fluvial deposits, and especially the planform geometry of rivers, is difficult. The sedimentary record of pre-Silurian fluvial systems generally displays monotonous and laterally extensive sand bodies (e.g., Davies & Gibling, 2010;Gibling, 2006;Gibling et al., 2014;Ielpi & Rainbird, 2016;Long, 1978Long, , 2011, with the mud content of alluvial successions sharply increasing near the Ordovician-Silurian boundary (McMahon & Davies, 2018a). In turn, preserved heterolithic lateral accretion sets, characteristic of some meandering rivers, only become common in outcrop-scale exposures near the Silurian-Devonian boundary (e.g, Cotter, 1978;Davies et al., 2011;Davies & Gibling, 2010;Gibling et al., 2014). Altogether, the clear spatial and temporal relationship between single-thread rivers and vegetation in both modern and ancient environments supports the hypothesis that land-plant evolution has influenced the development of fluvial systems through time and that bank-strengthening vegetation promotes the formation of singlethread rivers.
It remains unclear, however, whether the interpreted paucity of pre-Silurian single-thread rivers reflects a true scarcity of such rivers before the advent of land plants, a lack of refined facies models applicable to nonheterolithic meandering systems (Hartley et al., 2015, or selective preservation of sandier deposits in the absence of vegetation. There is also a possibility that some Precambrian fluvial mudstones were misinterpreted as marine in the absence of fossils (McMahon & Davies, 2018a). Despite being rarely recognized (e.g., Gibling et al., 2014;Hartley et al., 2018;McMahon & Davies, 2018b), at least some single-thread rivers existed long before plants colonized land (Long, 1978(Long, , 2011Ielpi & Rainbird, 2015;Santos & Owen, 2016;Ielpi et al., , 2017. Moreover, some single-thread rivers meander on Earth today in the virtual absence of macroscopic plant life (Matsubara et al., 2015;Ielpi, 2018;Ielpi & Lapôtre, 2019a-b;Santos et al., 2019). Finally, another barren endmember of single-thread rivers comes from Mars. Although its surface is largely cold and dry today, abundant evidence suggests that Mars once had an active surface hydrological cycle, with lakes, rivers, and deltas (Di Achille & Hynek, 2010;Fassett & Head, 2008;McLennan et al., 2019). In particular, many sinuous ridges on Mars are thought to be remnants of deposits from sinuous-tomeandering rivers where neighboring floodplain deposits have been deflated and channel fills or channelbelt deposits now stand in positive relief (Burr et al., 2010;Cardenas et al., 2018;DiBiase et al., 2013;Kite et al., 2015;Williams et al., 2013). It is thus clear that, even though land plants play a significant role in shaping single-thread streams on Earth today, they are not a necessary condition for single-thread rivers to form.
If bank strength is necessary to form single-thread rivers, strengthening agents other than plants must have played a role on the pre-Silurian Earth and ancient Mars; candidates include ground ice, cements, and cohesive mud. Even though single-thread rivers are found in terrestrial permafrost, they often have vegetated banks (e.g., Matsubara et al., 2015). Furthermore, ice has been shown to promote bank erosion where temperatures oscillate annually around the freezing point of water (Hinkel & Hurd, 2006;Lawler, 1986;Scott, 1978;Wolman, 1959), a necessary condition for liquid water to be stable within the channel while ground ice permeates riverbanks. Altogether, any relationship between ground ice and single-thread rivers remains unclear. Salts, carbonates, and phyllosilicates are abundant on Earth, have been detected by spectrometers orbiting Mars (e.g., Ehlmann et al., 2008;Ehlmann & Edwards, 2014;Mustard et al., 2008;Poulet et al., 2005), and may have provided cohesion to ancient terrestrial and Martian riverbanks. However, cementing chemical precipitates may dissolve by contact of flowing unsaturated water. The few terrestrial unvegetated rivers that are single threaded (and even meander) occur in mud-prone endorheic basins (e.g., Figure 1a; Matsubara et al., 2015;Ielpi, 2018;Ielpi & Lapôtre, 2019a, 2019b, leaving mud as the simplest explanation for the formation of single-thread rivers on ancient Earth and Mars.
Part of the issue in evaluating the importance of different bank strengthening mechanisms is that, even though sophisticated theoretical and numerical models have been developed to study the formation and dynamics of single-thread and meandering rivers (e.g., Crosato & Saleh, 2011;Eaton et al., 2004;Eaton et al., 2006;Howard & Knutson, 1984;Ikeda et al., 1981;Matsubara & Howard, 2014;Millar, 2000;Millar & Quick, 1993Parker, 1976;Parker et al., 2011), few studies have attempted to relate bank strength quantitatively to channel planform (Dunne & Jerolmack, 2018). Even studies that have attempted to relate river planform to bank properties parameterized bank strength through a single parameter that encompasses all physical, chemical, and biological contributions and requires field calibration for individual rivers (Eaton et al., 2004;Millar, 2000;Millar & Quick, 1993. Specifically, there is no physics-based model capable of predicting the formation of single-thread rivers within a wide range of riverbank materials and that does not systematically require field calibration of bank erodibility. To bridge this gap, we develop a theory for the formation of single-thread rivers in barren landscapes and evaluate it against observations of ancient fluvial deposits on Earth and Mars.

Goals
The overarching goal of this study is not to capture the complex dynamics of single-thread rivers but, rather, (i) to provide a first-order mechanistic model for the capacity of mud to confine water within a single channel (section 3) and (ii) to develop a new quantitative method to interpret the deposits of unvegetated rivers (section 4). We focus on mud (clays + silt) because it is ubiquitous on Earth today, in ancient (including pre-Silurian) sedimentary rocks (e.g., Tosca et al., 2010) and in ancient Martian terrains (e.g., Ehlmann & Edwards, 2014), and may be the primary bank-strengthening agent that allows for single-thread rivers in many environments (Dunne & Jerolmack, 2019). Determining whether mud alone can cause the formation of single-thread rivers is important because field observations of single-thread rivers in mud-prone environments is often confounded by the presence of cementing sulfates and putative biofilms on Earth (e.g., Ielpi, 2018) and because properly scaling flume experiments is challenging (e.g., Kleinhans et al., 2014;. Building on our model results, we discuss implications for quantifying the effect of vegetation on bank strength (section 4.1), the hydraulics of modern unvegetated single-thread rivers (section 4.2), paleohydraulic applications to pre-Silurian fluvial sedimentary rocks (section 4.3), and hydrologic and climate scenarios for Early Mars (section 4.4).

Model: Single-Thread Rivers Without Plants
We develop a simple model to relate a river's equilibrium width-to-depth ratio (W/h) to a set of quantities that are readily measurable in the field and then use W/h as a proxy for river planform geometry (Engelund & Skovgaard, 1973;Parker, 1976). Channel width is often difficult to quantify in the sedimentary record, given the rare occurrence of outcrops that are laterally continuous over the 100-to 1,000-m scale of most natural channels (Ghinassi et al., 2013) and a general lack of preserved stratal features that can be interpreted in terms of channel width (e.g., channel fills). In contrast, channel depth may readily be estimated from bedforms, bar cross set thicknesses (e.g., Bradley & Venditti, 2019;Mohrig et al., 2000), and channel-belt thicknesses (Hayden et al., 2019). In the following, we thus use flow depth, h, as a known input parameter to estimate W, although the model could easily be formulated to use W as an input and to output h.

Channel Equilibrium Width
The width of rivers in cohesionless substrates attain equilibrium when net bank erosion is null (e.g., Métivier et al., 2017;Parker, 1978). Similarly, the equilibrium width of alluvial rivers forming within cohesive banks must be such that, on average, net bank erosion is null (e.g., Millar & Quick, 1993Millar, 2000;Dunne & Jerolmack, 2019). Whereas such an equilibrium can either be static (and achieved through the total absence of erosion and deposition) or dynamic (and be achieved through any given amount of bank erosion being compensated by the same amount of deposition), either scenario requires the mean depth-averaged bank stress (τ bank ), as averaged over both inner and outer banks, to be close to the threshold for the erosion of bank materials (τ crit ; Dunne & Jerolmack, 2018), that is, τ bank ≈τ crit : (1)

Bank Stresses
Shear stresses exerted by water on the banks is a function of bed shear stress and channel cross-sectional shape (e.g., Flintham & Carling, 1988;Kean et al., 2009;Kean & Smith, 2004;Knight et al., 1984;Nelson & Seminara, 2011) and may be estimated through where ε is a stress-partitioning function and where total bed shear stress (τ bed ; skin friction + form drag) can be estimated from bed slope (S) and h assuming steady, uniform flow conditions (τ bank = τ bed = ρghS, with ρ the density of water and g the acceleration of gravity) for modern rivers, or from sedimentological constraints of bed regime for ancient deposits (e.g., Lapôtre et al., 2017). We employ the stress-partitioning function of Knight et al. (1984) and Flintham and Carling (1988) for a rectangular channel, assuming similar wall and bed roughness (Text S1), Intuitively, stresses partition such that τ bank~τbed for narrow and deep channels (W/h~1) and τ bank → 0 in the limit of very wide and shallow rivers (W/h → ∞). Even though ε was determined empirically, it is defined as the ratio of two shear stresses that have the same gravity dependence, such that it should be applicable to the Martian environment.

Critical Stress for Bank Erosion
The critical stress for bank erosion is a complex function of bank materials and fluid properties (Grabowski et al., 2011). Perhaps owing to this complexity, bank strength has most often been taken into account through a single parameter-an effective bank friction angle (e.g., Eaton et al., 2004;Millar, 2000;Millar & Quick, 1993) that encompasses all strengthening effects (e.g., grain contacts, mud cohesion, and vegetation). Rather than parameterizing bank resistance to erosion through a single parameter, here we explicitly model the effect of cohesive sediments on the bank erosion threshold, τ crit . We utilize the model of Ternat et al. (2008) to predict τ crit as a function of grain size within the banks (d bank ). Ternat et al. (2008) assume that cohesion arises for fine clay-to-silt-sized particles through van der Waals forces and neglects structural and double-layer electrostatic interactions. These assumptions are most valid for water-saturated materials that are not fully consolidated, as is expected in the active surface layer of riverbanks (see also Text S2). Whereas banks are modeled with a single effective grain size to limit the number of variables in this analysis, the model formulation has the capability to determine cohesion for a mixture of different grain sizes (Ternat et al., 2008). In dimensionless form, where ρ s is sediment density, τ * 0 is the critical Shields stress for the incipient motion of loose sediment, and F c and F w are the cohesive force and weight of particles, respectively. For relatively large particles, the critical stress for erosion is determined by the Shields stress for cohesionless sediment, whereas for small particles, the cohesive-force term dominates, and the critical stress for bank erosion becomes significantly greater than  Table 1). In (c), the predicted stability field of single-thread rivers is left unshaded. that for cohesionless particles (Figures 2a and 2b). The fit of Soulsby and Whitehouse (1997) to experimental data for the Shields stress of cohesionless grains is used, is a bank-specific particle Reynolds number, with R ¼ ρ s −ρ ρ being the submerged specific density of sediment, and ν is kinematic viscosity of the fluid (Text S2).
Particle weight is given by where k w is a shape factor equal to π 6 = for spherical particles. Under the assumption of a single grain size (Text S2), the cohesive force is calculated through where A h ≈ 10 −20 is the Hamaker constant, β is the coordination number of sediment grains, ϕ is a characteristic angle of internal friction, and K n is the compaction function given by with n the porosity and n c and n max are the fully compacted and maximum porosities, respectively (Ternat et al., 2008). Ternat et al. (2008) do not define ϕ as a friction angle in the Mohr-Coulomb sense (i.e., tanϕ≠ τ−c σn where τ and σ n are the shear and normal stresses, respectively, and cohesion, c, is accounted for in the numerator) but rather as the ratio of driving to stabilizing forces (i.e., tanϕ ¼

Fd
Fw −FlþFc where F d and F l are drag and lift forces, respectively, and where the cohesive force, F c , is accounted for in the denominator); hence, it has a higher value than those typically reported for granular materials in geotechnical studies. Rewriting τ * crit as a function of Re p,bank using equations (4)- (8), we express the model of Ternat et al. (2008) as where Ω ¼ Ahβ 1−cosϕ ð Þ 48K 2 n kw ρν 2 dbank is a dimensionless number describing the ratio of energies associated with van der Waals interactions (and thus a function of sediment compaction through β and K n ) and grain-scale viscous energy. Assuming hexagonal close packing of sediment grains, we set β = 12 and n c ¼ 1− π 3 ffiffi 2 p ≈0:26, and following Ternat et al. (2008), we set ϕ = 52.5°and n max = 1. These values yield accurate results for natural granular materials (Ternat et al., 2008).

Cohesion of Vegetated Banks
Previous studies have quantified the effect of vegetation on bank strength for specific rivers or particular plant species (e.g., Micheli et al., 2004;Micheli & Kirchner, 2002;Polvi et al., 2014;Simon & Collison, 2002). Here we take a simpler, more general approach and include vegetation into equation (4) with a single parameter, where σ * is a dimensionless strength factor describing vegetation-added strength and is defined as Þ gdbank and τ veg being the additional fluid stress to erode banks required by the presence of 10.1029/2019JF005156 Journal of Geophysical Research: Earth Surface vegetation. We do not know the functional form of τ * veg , so we use equation (10) only to determine the value of σ * that minimizes the discrepancy between data from vegetated rivers and model predictions.

Formation of Single-Thread Channels
By substituting equations (2) and (4) into equation (1), we solve for river equilibrium width (W) and W/h as a function of eight main input variables-h, S (or shear velocity, u * ¼ ), d bank , g, ρ, ρ s , ν, and n (we keep all other parameters constant). Our approach is similar to that of Millar andQuick (1993, 1998), Millar (2000), and Eaton et al. (2004) but with two main differences: (i) Bank cohesion in our model is calculated as a function of grain size and other properties of bank sediments using an explicit first-principle model, and (ii) the model does not require an optimization condition on bed slope, sediment transport, or flow resistance because channels are assumed to be rectangular (Bui et al., 2000). Parker (1976) showed that W * , the maximum W/h achieved by single-thread rivers at the meandering-tobraiding transition, depends on bed slope (i.e., W * ¼ Fr Fr is the flow Froude number and C f is a dimensionless friction coefficient) but not on gravity (at least to first order; gravity may affect, e.g., bedforms, and thus C f ). According to Parker (1976), rivers develop midchannel bars and braid at W * ≈ 100 − 200 (Text S3). Given the dependence of W * on channel slope and to account for the wide range in bed slope spanned by the natural and experimental rivers we next investigate, we use three different W * values that follow a W * ∝ 1 ffiffi S p dependence. We employ a conservative value of W * = 200 to illustrate the braiding-to-single-thread transition for S < 10 −3 , W * = 65 for S~10 −3 − 10 −2 , and W * = 20 for S~10 −2 − 10 −1 . Because W * is not expected to vary significantly with gravity, we use the same W * values for Earth and Mars. The stress-partitioning model we use only applies to single-thread rivers; thus, while our

10.1029/2019JF005156
Journal of Geophysical Research: Earth Surface model can predict the single-thread-to-braiding transition, it does not apply in the braided regime. Finally, we find that, for siliciclastic grains in freshwater, cohesionless bank materials approach their threshold for incipient motion as W/h approaches a value of 3 (e.g., Figures 3 and 4). Because bank materials need to be transported in the first place to be deposited overbank, we adopt W/h = 3 as a conservative lower limit for the formation of single-thread rivers.

General Predictions
For cohesionless sediments, the model of Ternat et al. (2008) shows that the fluid stress required to mobilize grains (τ 0 ¼ τ * 0 ρ s −ρ ð Þgd bank ) generally decreases as d bank decreases and follows two different asymptotic behaviors in the hydraulically smooth (Re p,bank <~10 0 ) and rough (Re p,bank >~10 2 ) regimes (Figures 2a and 2b). In contrast, the critical stress for erosion of cohesive sediments (τ crit ) increases as d bank decreases when cohesion becomes significant (i.e., for mud-sized particles and thus in the smooth regime), such that the critical stress for erosion may be described through three asymptotic regimes as grain size increases-the cohesive-smooth (Re p,bank < Re p,c , where typically Re p,c <~10 0 ), transitional (Re p,c < Re p,bank <~10 2 ), and cohesionless-rough (Re p,bank >~10 2 ) regimes (Figures 2a and 2b). Because van der Waals forces are a function of the distance between grains, the grain size at which  Table 1 summarizes model input parameters for each panel and Table 2 Table 1).
Under steady uniform flow and at constant h, τ bed (and thus τ bank ) is a linear function of bed slope, S. As such, under our model assumptions, an equilibrium river of fixed W/h (i.e., for which τ bank = τ crit ) follows three regimes with increasing d bank -first, S decreases with increasing d bank in the cohesive-smooth regime; second, S increases with coarser bank materials as cohesion becomes negligible in the transitional regime; and third, S increases more rapidly with d bank in the cohesionless-rough regime (Figures 2c). In Figure 2c, contours show predicted W/h, and the 3 < W/h < 200 envelope illustrates the stability field of single-thread rivers plotted for typical terrestrial conditions (Table 1; see also Text S4 for a sensitivity analysis). A first important prediction is that single-thread rivers can readily form with fine-grained banks in the absence of bank vegetation. The lower bound for the formation of single-thread rivers corresponds to conditions near the threshold of motion for bank materials when cohesion is negligible because channels with W/h~1 have τ bank~τbed (and by construction τ bank = τ crit ). The majority of the single-thread-river stability field is at bed slopes below the threshold of suspension of the bank materials, except for (i) silt-to-clay-sized particles, which are easily suspended because they are light, and (ii) a narrow grain-size range within the smoothto-rough transition (with respect to Re p,bank ; medium sand for siliciclastic grains in ambient freshwater on Earth). Furthermore, the stability field of single-thread rivers is expected to shift to steeper slopes for shallower flows (e.g., Figures 4a vs. 4c), because at constant W, shallower flows require steeper slopes to achieve τ bank = τ crit (d bank ).

Single-Thread and Braided Rivers in Flume Experiments
Here we evaluate the model against results from flume experiments that produced braided and single-thread rivers (Table 2) , however, were able to produce repeated chute cutoffs. Braudrick et al. (2009) also produced singlethread meandering rivers with W/h~30 by feeding a mixture of 800-μm sand and 300-μm lightweight plastic particles to the flume and by seeding the floodplain with alfalfa sprouts that added bank strength (Figure 3b).  (Figure 3), Terrestrial Rivers (Figure 4), and Martian Rivers (Figures 6 and S2  . Consistent with the model, these experiments fall outside of the predicted stability field of single-thread channels (Figures 3c  and 3d).

Terrestrial Rivers Forming Within Muddy Unvegetated Banks
Since Grain sizes in riverbanks of the Amargosa River, Bonneville Basin, and Toiyabe Basin were qualitatively constrained by Ielpi et al. (2018) and Ielpi andLapôtre (2019a, 2019b) and consist of clay and silt with lenses of very fine to fine sand, similar to Lake Lahontan sediments forming the banks of the Quinn River, NV (Matsubara et al., 2015). The model correctly predicts that the Quinn River, NV, should be single threaded.
The model further predicts that both the Amargosa River and washes of the Bonneville Basin should be single threaded, given observed bed slopes, if the effective bank grain size is in the clay-silt range. These effective bank grain sizes are qualitatively consistent with onsite observations and quantitatively consistent with Lake Lahontan sediments incised by the Quinn River in a similar depositional setting.

Quantifying the Effect of Bank Vegetation
Numerical and theoretical models are often used to evaluate the effect of vegetation on bank strength and stream evolution within vegetated banks (e.g., Camporeale et al., 2013;Crosato & Saleh, 2011;Eaton & Giles, 2009;Millar, 2000;Murray & Paola, 2003). Because the theoretical model presented here makes predictions for the formation of single-thread rivers in the absence of vegetation, one can in principle separate the strengthening effects of vegetation on riverbanks from those of sediment properties and quantify them by comparing model predictions with data from vegetated rivers. Specifically, σ * (equation (10)) can be used to minimize the misfit between predicted and observed W/h.

Note.
Values in parentheses indicate those used for error bars.
a Reported values are overall maximum thalweg depths, such that average channel depth in deepest reaches is estimated to half of the reported values, and overall average depth is estimated to be shallower by a few millimeters (consistent with description of Peakall et al., 2007

Journal of Geophysical Research: Earth Surface
(shaded green field in Figure 3b) match experimental data. We conduct a similar exercise for natural vegetated single-thread rivers using the compilation of Dong et al. (2019), which includes data from singlethreaded reaches of the Selenga River delta (Russia), a set of gravel-bedded rivers from England, the Llano River (United States), the Fly River (Papua New Guinea), and the Siret River (Hungary). Channel reaches from Dong et al. (2019) have W/h~4-136, S~2 × 10 −5 − 2 × 10 −2 , and d bank~5 -465 μm. The compiled data from vegetated rivers plot within the braided-stability zone of the (d bank , S) space ( Figure 4) and are predicted to have W/h > 200 in the absence of vegetation, indicating that vegetation is likely an important bank-strengthening agent for these rivers. For example, the banks of the deeper rivers (h = 9-11 m; Figure 4c) would need to be composed of clay to fine silt to match model predictions in the absence of vegetation; yet their banks consist of coarse silt to fine sand. An added strength of σ * = 29 due to vegetation allows model predictions to match observations (Figure 4d), consistent with the strengthening effect of riparian shrubs and trees (Polvi et al., 2014;Smith, 1976).

Hydraulic Geometry of Single-Thread Rivers Forming Within Muddy Unvegetated Banks
Terrestrial rivers were shown to adjust their bankfull geometry such that their formative Shields  (11) (pink dashed line in Figure 4; Parker et al., 2007;Wilkerson & Parker, 2010;Trampush et al., 2014). This empirical relationship predicts relatively accurately the bankfull geometry of vegetated rivers (Figure 4). Similar empirical relationships are often used to infer paleohydraulic conditions from fluvial deposits on Earth (e.g., Mahon & McElroy, 2018) and Mars (e.g., Jacobsen & Burr, 2016). However, the degree to which such relationships may be applied to unvegetated systems, such as pre-Silurian (e.g., Eriksson et al., 2006;Mukhopadhyay et al., 2014) or Martian (e.g.,  rivers, is not well understood. Moreover, bank cohesion is not accounted for under this choice of dimensionless variables, and significant scatter remains around the derived relationship (e.g., Wilkerson & Parker, 2010).
Single-thread rivers forming within muddy unvegetated banks (Figures 4a and 4b) are good systems to test the importance of bank cohesion on bankfull geometry. The compiled rivers (Amargosa River, Quinn River, washes of the Bonneville Basin, and streams of the Toiyabe Basin) have fine grains on their beds (silt to medium sand), such that their formative Shields stresses fall within the scatter of vegetated rivers (pink crosses in Figures 4a and 4b). Therefore, unvegetated and vegetated rivers on Earth seem to have statistically similar hydraulic geometries (Ielpi et al., 2017), and terrestrial unvegetated single-thread rivers might be described well by equation (11). For rivers with cohesive banks, bankfull geometry is perhaps more intuitively described using a bank-specific Shields stress (τ *;bank e hS Rdbank ) and a bank-specific particle Reynolds number (Re p;bank ¼ ; e.g., Li et al., 2015). However, in the cohesive limit, the formative τ *,bank is not solely a function of Re p,bank , and another dimensionless number (Ω) is required to describe cohesion (equation (9)). In the cohesive limit, τ *;bank ∝ΩRe −2 p;bank , such that the equilibrium bed slope for a given W and h is expected to be proportional to d −3 bank in the cohesive regime (e.g., Figure 4); conversely, at constant S, W/h is expected to increase with increasing d bank . The latter prediction is qualitatively consistent with the analysis of the vegetated Selenga River delta of Dong et al. (2019). In summary, the equilibrium bankfull geometry of single-thread rivers forming in unvegetated cohesive sediments is expected to be a strong function of bank grain size.

Applications to Pre-Silurian River Deposits
Few single-thread river systems have been reported relative to braided systems in the pre-Silurian sedimentary record based on the presence of heterolithic lateral accretion sets (e.g., Gibling et al., 2014).

Journal of Geophysical Research: Earth Surface
retaining mud fractions from oceanward transport (e.g., Dott, 2003) should also have formed before the rise of land plants and provided favorable geodynamic and paleoclimatic settings for the accumulation of mud (Nichols, 2012). In a compilation of prevegetation fluvial floodplain deposits, Ielpi et al. (2018) envisaged a causal link between the rise of supercontinental assemblages, the thorough chemical weathering of large orogenic belts therein, and the deposition of mud-rich strata in low-gradient, endorheic-prone terrestrial basins. Moreover, analyses of fine-grained Proterozoic sedimentary rocks show that both detrital and pedogenic clays were produced abundantly more than a billion years before the rise of land plants (Tosca et al., 2010). Thus, despite a reported paucity of mudrocks in the identified pre-Silurian fluvial record, there is no a priori reason why single-thread rivers should have been rare on the pre-Silurian Earth. In fact, several single-thread river deposits have been identified in strata that predate the greening of the continents based on detailed observations of point bars and channel fills (Long, 1978(Long, , 2011Ielpi & Rainbird, 2015;Santos & Owen, 2016;Ielpi et al., , 2017 as well as dune cross stratification (e.g., Ganti et al., 2019).
Combining our model predictions with a dune-stability criterion (Lapôtre et al., 2017), we show that riverbanks need strength for dunes to form in coarse-sand or finer beds (Text S5 and Figure S1), consistent with the recent analysis of Ganti et al. (2019) of prevegetation fluvial deposits of the Torridon Group of Scotland. The question remains, however, of what strengthened pre-Silurian riverbanks. Mud contents as low as 3% by weight cause sand/mud mixtures to become cohesive, with cohesion increasing nearly linearly with mud content up to~20-30% mud (Mitchener & Torfs, 1996;Ternat et al., 2008). It is thus possible that some of the observed sandy deposits are in fact floodplain deposits, and small amounts of mud were either removed or overlooked. This idea is consistent with recent observations of a modern point-bar deposit along a highly mobile unvegetated river meander in the Toiyabe Basin, NV (Ielpi & Lapôtre, 2019b), which is dominated by sand-sized materials with little intercalated mud. If found to be representative of other unvegetated meandering streams, the observed stratigraphy in the Toiyabe Basin is inconsistent with heterolithic lateral accretion sets being diagnostic of river meandering (as also suggested by, e.g., Hartley et al., 2018;McMahon & Davies, 2018b;Swan et al., 2018).
In the absence of clear heterolithic lateral accretion sets, the deposits of pre-Silurian rivers may preserve indirect clues about bank materials and channel width that our model can help decipher ( Figure 5). First, one may use our model to infer the effective grain size of bank materials based on observations of channel deposits. In the case of polydisperse riverbanks, the effective grain size inferred through our model would

Journal of Geophysical Research: Earth Surface
reflect the sediment size that would yield the same bank strength as the true banks. Thus, an inverted grain size in the clay-to-silt range would suggest that banks were eroding cohesively and thus that some amount of mud (>3%; Mitchener & Torfs, 1996;Ternat et al., 2008) was likely present. For example, Ganti et al. (2019) independently determined that rivers that formed the Upper Applecross Formation of the Torridon Group in Scotland had typical depths of~10 m and bed slopes of~10 −4 . They also inferred that those streams were single threaded. Our model suggests that such deposits can be explained by flows confined within banks composed of sediments with an effective grain size of~3-8 μm or in the clay-to-silt range and thus that mud could have strengthened the riverbanks (Figure 4c). Although fine-grained floodplain deposits are not abundant, pre-Silurian alluvial strata typically contain a few percent mudrock (McMahon & Davies, 2018a), and the grain size of fines in putative bank-forming materials can be estimated from fine-grained intervals or lenses in the otherwise sand-dominated deposits. Using the full formulation of Ternat et al. (2008) for polydisperse sediments and observations of fine-grained intervals in the Torridon deposits, one could quantitatively determine the amount of mud required to produce the bank strength equivalent to 3to 8-μm grains.
Second, it is difficult to determine channel width in the absence of floodplain sediments in pre-Silurian fluvial strata. Flow depth and bed stress can be characterized by inspection of bedforms and barforms within channel deposits and used in a stress-partitioning model to estimate bank stresses (equations (2) and (3)). Bank-strengthening grain sizes can be estimated by inspection of fine intervals and fed into a bank-cohesion model (equations (4)- (8)). Paleo-channel width can then be estimated by iteratively solving equation (1), using the inferred W/h combined with estimates of paleo-bed slope (e.g., from bed stress estimates, bed grain size, and flow depth; Lynds et al., 2014;Trampush et al., 2014;Mahon & McElroy, 2018; Figure 5). For example, consider a putative ancient channel-bed deposit from an Amargosa-like river (Ielpi, 2018), with grain sizes of~100 μm and that contains~1-m-tall bar forms and current ripples with wavelengths of~11 cm. Using the formulation of Lapôtre et al. (2017), we determine that the formation of 11-cm ripples in 100-μm grains required bed stresses of τ b~0 .6 Pa. Further assuming that bar height can be used as a 1:1 proxy for flow depth, h, bed slope can be estimated under steady uniform flow conditions as S ¼ τb ρgh ≈6×10 −5 . The ancient Amargosa-like deposit also contains homogeneously fine-grained intervals in the form of mud drapes. Assuming that they are representative of sediments that provided strength to the riverbanks, bank grain size can be estimated by inspection of those fine intervals. For a bank grain size of~10 μm, our model would predict that the river that formed the observed deposits had a W/h ≈ 20 or a channel width of~20 m ( Figure 4a). If instead the finest intervals had a grain size of~50 μm, our model would predict an equilibrium W/h > 200, indicating the rocks were deposited by a braided river (Figure 4a).

Single-Thread Rivers on Early Mars and Implications for Climate Scenarios
In order to illustrate how the ancient Martian environment may have influenced the formation of singlethread rivers, predicted W/h are shown for a shallow Martian river (h = 1 m), where only gravity and the density of sediments (quartz-dominated on Earth vs. basaltic on Mars) were varied relative to their terrestrial equivalents ( Figure 6a and Table 1; see also Text S6 and Figure S2). Although Martian clays are dominated by Fe/Mg rich minerals (e.g., Ehlmann & Edwards, 2014;Mustard et al., 2008;Poulet et al., 2005), the effect of clay mineralogy is ignored here (Text S2). Compared to terrestrial rivers, Martian single-thread rivers are predicted to form at steeper bed slopes (Figure 6a). Intuitively, the lower Martian gravity implies lower bed stresses at a given slope, such that achieving similar bank stresses on both planets requires steeper riverbeds on Mars. In addition, the degree to which a Martian river must steepen to achieve Earth-like bank stresses is not the same for all bank materials. By Taylor expansion of τ * crit Re p;bank À Á (equations (5) and (9)), we find that for a given grain size, Martian rivers (relative to their terrestrial counterparts) are expected to have slopes e g E g M ≈2:6 times steeper in the smooth-cohesive limit (because of the lower Martian gravity) and e RM RE ≈1:2 times steeper in the cohesionless-rough limit (because of the higher density of basaltic sediments) but to have similar slopes in the transitional regime (where subscripts "E" and "M" indicate Earth and Mars, respectively; Figure 6).
Shifting the stability domain of single-thread rivers to steeper equilibrium slopes is equivalent to forming narrower rivers on Mars at a given bed slope. Figure 6b shows the ratio of predicted Martian-to-10.1029/2019JF005156

Journal of Geophysical Research: Earth Surface
terrestrial channel widths ( W M W E ) as a function of d bank and S. Consistent with the previous analysis, at constant d bank and S, Martian rivers are found to be significantly narrower than their terrestrial counterparts ( W M W E ≈0:1) in the smooth-cohesive regime, whereas Martian rivers have similar equilibrium widths in the transitional regime and again become moderately narrower than terrestrial rivers in the cohesionless-rough regime ( W M W E ≈0:6 within fine-gravel banks; Figure 6b). Our model predictions are qualitatively consistent with the inference of Konsoer et al. (2018) that Martian rivers must have steeper beds than terrestrial rivers of similar W/h. Specifically, Konsoer et al. (2018) proposed empirical relationships to predict bankfull depth (h) and flow discharge (Q 0 ) from channel width on Mars, and Q 0 ¼ 0:2W 1:68 : Assuming steady uniform flow and a constant dimensionless bed-friction factor (C f ), bankfull discharge for a rectangular channel may be rewritten as such that bed slope can be found as a function of W by combining equations (12)-(14) as Provided that W can be estimated from orbiter-based imagery, equations (12) and (15) can be used to determine W/h and S, which then can be combined with our model to determine d bank . For example, Kite et al. (2015) measured ridge widths between~10 and~50 m at Aeolis Dorsa and assumed these were equivalent to channel widths. Combined with equation (12), the estimates of Kite et al. (2015) imply channel depths of   (Wright & Parker, 2004), equation (15) constrains channel paleo-slopes to be~4.3 × 10 −4 − 1.2 × 10 −3 at Aeolis Dorsa, in agreement with the independent estimates of 5 × 10 −4 − 1 × 10 −3 from DiBiase et al. (2013) for ridges in the same area. Using these values, our model predicts that Aeolis Dorsa rivers must have flowed within banks whose eroding behavior was equivalent to that of either fine silt or fine gravel (illustrated in Figure 6a for representative values of h~1 m and W~15 m and with S~9 × 10 −4 ). Fine-grained banks are more consistent with the now wind-eroded floodplains that led to ridge formation.
Even though floodplain deposits were preferentially eroded to form Martian sinuous ridges, detrital cohesive fine-grained sediments have been observed by the National Aeronautics and Space Administration's (NASA) Curiosity rover within Gale crater (e.g., Bristow et al., 2018;Grotzinger et al., 2015;Schieber et al., 2017). In addition, the global abundance of clay minerals as detected by orbiter-based spectrometers in Late-Noachian-to-Early-Hesperian terrains (Ehlmann & Edwards, 2014;Mustard et al., 2008;Poulet et al., 2005) attests to significant clay production in the early history of Mars. Provided observations of bank-materials grain sizes from Martian river deposits, possibly as early as the 2020s when NASA's next rover will investigate a river delta deposit within Jezero crater (e.g., Goudge et al., 2017;Schon et al., 2012), our model can be used to evaluate whether mud in riverbanks can explain the origin of singlethread rivers and as a paleohydraulic tool to decipher ancient Martian fluvial deposits (Text S6 and Figure S2). Determining the bank materials of ancient Martian rivers is important for the hydrologic and climate history of Early Mars-clays imply silicate weathering which in turn could indicate warmer surface temperatures during channel-forming episodes (e.g., Bishop et al., 2018;Carter et al., 2015) relative to an ice-cemented, mud-free scenario, which instead would imply near-freezing surface conditions with only limited silicate weathering at the surface. Alternatively, chemical cements would likely imply a net evaporative surface environment.

Conclusions
Single-thread river systems are a conspicuous part of Earth's land surface today. A strong spatiotemporal correlation between the occurrence of single-thread rivers and vegetation supports the hypothesis that land-plant evolution has influenced the development of fluviatile systems through time. Despite this correlation, a few single-thread-river deposits have been recognized in the fluvial rock record prior to the advent of land plants, and meandering-river deposits are found in vegetation-free basins on Earth and on Mars, where macroscopic plants most likely never evolved. Thus, although vegetation may have played an important role in shaping Earth's rivers, rivers do not require vegetation to be single threaded. Here we explore whether cohesive mud may provide riverbanks with sufficient resistance to erosion for flows to be confined within relatively deep and narrow channels without vegetation. We combine equations of open-channel flow hydraulics with a physics-based erosion model that is applicable to a broad range of bank sediments. The new model permits us to make predictions of river width-to-depth ratio (W/h), which correlates with its planform geometry. The model predicts that single-thread rivers can form with a broad range of bank substrates, from clay to gravel banks, consistent with rivers formed in flume experiments and within largely unvegetated mud-prone endorheic basins of the western United States.
The model has several useful applications. First, we show how the model can be used to infer the strengthening effect of vegetation on riverbanks by quantifying the contribution of mud cohesion. Second, we discuss possible controls on bankfull hydraulic geometry of single-thread rivers with muddy unvegetated banks and show that width should be a strong function of bank grain size. Third, in application to pre-Silurian fluvial deposits, we propose a workflow to determine bank strength (and thus the effective grain size of bank materials) or reconstruct a river's planform geometry from a simple set of field observables (grain size, bedforms, etc.). This workflow holds promise in deciphering indirect clues from the pre-Silurian fluvial record. Finally, we show that single-thread rivers are predicted to form within a broad range of bank materials on Mars.
Owing to lower gravity, Martian rivers are expected to be narrower than their terrestrial counterparts at a given slope. Conversely, rivers of a given width are expected to have steeper bed slopes on Mars in order to achieve the necessary bank stresses to cause bank erosion. Future in situ observations of Martian 10.1029/2019JF005156

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single-thread-river deposits can be used with our model to constrain whether sufficient bank strength was provided by mud. If not, chemical cements or ground ice may have played an important role in forming meandering rivers on Early Mars.