Similarity solutions for fluvial sediment fining by selective deposition



[1] Current models for downstream sediment sorting by selective deposition generally perform well at describing observed sorting data. However, since most were developed initially for application to modern rivers, they are typically formulated in terms of hydraulic and bed-surface variables that are not readily measurable in the sedimentary record. Moreover, their algebraic complexity obscures some of the underlying simplicity of the segregation process. Here we show how a pair of hydraulically based sorting models developed by Parker et al. can be reformulated, with minimal loss of accuracy, in terms of the size distribution of the supplied sediment and the downstream depositional mass balance. By invoking constant dimensionless shear stress within either the gravel or sand regimes, reach-scale, short-term details of hydraulics and sediment transport are summarized via a pair of dimensionless relative mobility functions, one for gravel and one for sand. Our approach yields simplified similarity solutions in which the long-term longitudinal grain-size distribution of the substrate and the relative mobility functions can be collapsed into self-similar forms in which only local mean and standard deviation of sizes in transport are used as scaling parameters. The formulation we propose offers a simple means to explore the impact of controlling variables on fining profiles and can be easily incorporated in long-term, basin-scale numerical stratigraphic models, avoiding the necessity of modeling the details of hydraulics and sediment transport. The model involves a minimum number of physically based parameters, the numerical values of which can be determined from the spatial distribution of rate of deposition, dimensionless shear stress, and the coefficient of variation of the supply gravel or sand size distributions.

1. Downstream Sediment Fining and the Concept of Similarity: Overview

[2] A common geomorphic observation in fluvial systems is their ability to sort sediments via several physical mechanisms. In particular, the tendency of bed material to become finer downstream is a critical property of aggrading rivers that must be accounted for when modeling fluvial systems since it is a primary driver of downstream changes in river planform and depositional facies [Leopold and Wolman, 1957; Heller and Paola, 1992; Paola et al., 1992a; Paola, 2000]. The two most common explanations for fluvial downstream fining are (1) abrasion, a mechanism in which large particles break down into smaller sizes by fracturing and attrition; and (2) selective deposition, a process that can be viewed as a kind of hydraulically driven sediment fractionation [Paola, 1988; Parker, 1991; Paola et al., 1992a; Ferguson et al., 1996; Rice, 1999]. The observations that fining rates are strongly positively correlated with deposition lengths, and that observed fining rates in natural depositional streams are often orders of magnitude higher than those that appear possible by abrasion alone, indicate that selective deposition is the dominant factor that causes downstream fining in most aggrading fluvial systems [Paola, 1988; Parker, 1991; Heller and Paola, 1992; Paola et al., 1992a; Paola and Seal, 1995; Ferguson et al., 1996; Cui et al., 1996; Robinson and Slingerland, 1998; Rice, 1999; Hoey and Bluck, 1999; Paola, 2000].

[3] The fine-scale physics of grain segregation is extremely complex, and this is reflected in the complexity and necessarily semi-empirical nature of existing treatments of the problem [Tetzlaff and Harbaugh, 1989; Snow and Slingerland, 1987; Flemings and Jordan, 1990; Slingerland et al., 1994; Robinson and Slingerland, 1998]. Most of the work on fluvial sorting involves detailed numerical formulations developed for the channel-reach scale. These include empirical or semi-empirical relations for the fractional transport and differential mobility in a mixture of sediment sizes [Deigaard and Fredsoe, 1978; Snow and Slingerland, 1987; Parker, 1991; Bridge and Bennett, 1992; Van Niekerk et al., 1992; Hoey and Ferguson, 1994; Slingerland et al., 1994; Paola and Seal, 1995; Robinson and Slingerland, 1998; Wilcock and Kenworthy, 2002].

[4] Given the necessary input data, the current models do a reasonably good job of accounting for available field and laboratory data on downstream size segregation. Stepping back from the details of predicting downstream sorting in a specific modern stream, however, leads us to ask if the sorting problem might be amenable to simplification for stratigraphic time and space scales. The rate of downstream fining in a depositional fluvial system fundamentally depends on the following three things: (1) the range of sizes in the sediment supply (there can be no fining if the supply is too well sorted); (2) the extent and spatial distribution of deposition in the fluvial system (selective deposition cannot operate without deposition); and (3) hydraulic, topographic, and grain-interaction effects that result in relatively greater mobility of finer grain sizes (see the work of Solari and Parker [2000] for an interesting exception). Although the first two effects are implicitly represented in typical sorting models, the main emphasis has been on factor 3. This is appropriate for applications to modern rivers at reach scales: detailed information about the hydraulics, local topography, and surface and subsurface grain size is readily available; and it is usually much harder to determine the size distribution supplied to a given reach or the fraction of the total supply extracted in the reach.

[5] A fluvial-stratigraphic perspective leads us to analyze the problem using a different approach. On the scale of a whole basin, the fluvial system can account for a significant amount, in the range of 30–70%, of the total sediment mass loss to deposition [Goodbred and Kuehl, 1998; Walling, 1999; Lauer and Parker, 2004]. Preserved strata, with careful interpretation, can provide a time-integrated picture of the spatial pattern of mass extraction, and the deposit in its entirety reflects the size distribution of supplied material (provided abrasion is negligible). On the other hand, hydraulic information is limited to that obtainable by paleohydraulic reconstruction, topography is only partially preserved, and (in our experience) bed surface layers are rarely preserved. On the basis of these observations, our main purpose here is to investigate the extent to which downstream size sorting can be modeled in terms of parameters that are readily measurable in the stratigraphic record. We also believe that there exists an underlying simplicity (at least for the long term) in what has so far appeared to be a complex problem requiring high levels of empiricism. For example, below we will present evidence that rivers tend to organize themselves in such way as to limit the variability of a major hydraulic factor in sorting, the boundary shear stress.

[6] Our approach is to recast existing sorting models into a modified form aimed at application to stratigraphic problems and intended to be as simple and analytical as possible. We use the concept of similarity to obtain analytical solutions, inspired by previously reported numerical model results, experimental observations and some limited field data that suggest that, under certain conditions of steady or near-steady state (for example, constant base level), rivers tend to develop self-similar bed profiles and self-similar substrate grain-size distributions [Paola et al., 1992b; Cui et al., 1996; Toro-Escobar et al., 1996; Seal et al., 1997].

[7] We emphasize the concept of similarity because of its power both in collapsing empirical data and in theoretical analyses (for example, reduction of the order of differential equations). Similarity transformations allow one to collapse many different curves into a single one, provided the appropriate scalings, commonly functions of the independent local variables, are used to provide the appropriate nondimensionalization. For example, Toro-Escobar et al. [1996] have shown that their experimental data on channel bed elevations η(x, t) (here x is downstream coordinate and t is time) observed at different times during the evolution of an aggrading fluvial deposit with constant base level can be collapsed onto a single long profile by using the total length of the fluvial system L(t) and the elevations of the feed point and shoreline, η(x = 0; t) and η(x = L(t); t), respectively, as the scaling factors. Thus, in this last example, the requirement of self-similar bed profiles for two arbitrary times, t1 and t2, can be written in the following form:

equation image

[8] Expression (1) above indicates that different long profiles of the entire fluvial system differ only by a scale factor of time. This can be interpreted in the following way: once dynamic equilibrium is attained by the fluvial system, then the time evolution of the long profile can be represented by a simple deformation (stretching) of the profile, where the amount of stretching is a simple algebraic function of time obtained using the scaling variables.

[9] The experimental data reported in the works of Paola et al. [1992b], Toro-Escobar et al. [1996], and Seal et al. [1997] also suggest that, aside from bed profiles, other parameters of the general problem of downstream fining due to selective transport, such as the substrate grain-size distribution, could also be collapsed into self-similar forms. However, in these additional examples, complete analyses of similarity and their corresponding scaling factors have not been reported yet. Finally, additional support for the extension of the similarity concept from bed profiles to substrate composition comes from the close connection between long profiles and substrate fining profiles, which has been pointed out repeatedly in the literature [Hoey and Ferguson, 1997; Hoey and Bluck, 1999; Paola, 2000].

[10] On the basis of these previous observations, our approach in this paper is to extend the concept of similarity to fining profiles and develop an analytical solution for steady long-term downstream sorting and substrate size distribution. Our specific goal is to offer a simplified modeling approach that requires only the minimal input information that one would need to model a fluvial depositional system (for example, source grain-size distribution, subsidence rate, and spatial distribution of deposition). We use similarity methods to summarize the main fluid and grain physics embodied in current sorting models, leading to new, simplified forms that can be readily included in models for the long-term evolution of alluvial deposits and stratigraphy.

[11] The problem of downstream fining amounts to partitioning the total variance in the sediment supply between local (at a site) variance (the “sorting” one would observe in a single outcrop, whether in a hand sample or among different beds) and the variance down the whole system, which is manifested as downstream fining [Paola and Seal, 1995]. If the input size distribution were uniform enough that hydraulic processes permitted all sizes to be deposited at each site in the same proportion as in the supply, then there would be local size variability but no downstream fining. On the other hand, if local hydraulic selectivity were perfect, there would be strong downstream fining but no local size variability. So, for a given input size distribution, the role of hydraulic processes is to set the local selectivity (the range of sizes that can be deposited together at one location). In our formulation, the local selectivity is set by a “relative mobility function” that embodies the details of hydraulics (effects of topography, variable shear stress, etc.) and sediment transport, including the particularities of sorting mechanisms. An important assumption is that the relative mobility function can be expressed in terms of the same similarity variable as the substrate size distribution. The goal is that the relative mobility function be as general as possible, but one might expect it to differ for different modes of transport such as bed load or mixed bed load suspension.

[12] In the next section, we show that this can be done using self-similar forms that depend upon the general size category of the sediment (i.e., sand or gravel) but not on the detailed particle dynamics or hydraulics. We propose general forms of the relative mobility functions on the basis of numerical experiments using detailed hydraulic models that were developed for the cases of gravel rivers [Parker, 1991] and sand-bed rivers [Wright, 2003]. Our theoretical results and key assumptions are also tested against predictions of these detailed hydraulic numerical models, as well as some available field and laboratory data.

2. Self-Similar Solution for the Final Substrate Grain-Size Distribution

[13] The governing equation we use for the problem of downstream fining is given here first, followed by our proposed similarity solution. Our starting point is a simplified version of a sorting model introduced first by Hirano [1971]. Hirano's model summarizes flow and sediment transport processes by using the following three layers: a transport (or bed load) layer, a bed-surface (or active) layer, and the lower substrate. In the model, sediment is exchanged between bed load and active layer continuously, even during conditions of equilibrium transport. On the other hand, sediment exchanges between the active layer and substrate when the bed aggrades or degrades. In the particular case of bed aggradation, material from the active surface layer passes to storage in the substrate, and therefore a so-called transfer function is required in order to predict the composition of the substrate, given transport and flow conditions [see Toro-Escobar et al., 1996]. Several forms for the transfer function have been suggested by different authors [Hirano, 1971; Parker, 1991; Toro-Escobar et al., 1996]. On long timescales, we will consider that the fine-scale dynamics in the sediment transfer between transport and permanent deposit is embedded in a function that we call the “relative mobility function”.

[14] The full governing equation, the fractional Exner sediment mass balance per unit width in a given cross-section of a channel, as given by the three-layer Hirano model, is [see Toro-Escobar et al., 1996]:

equation image

where f is the fraction of a given sediment size in the substrate immediately below the bed-surface active layer, F is the proportion of the same size present in the surface layer, and p is the fraction of that size in the mixture in transport; η is the elevation of the channel bed referenced to an arbitrary fixed datum, La is the thickness of the surface (active) layer, customarily taken equal to few grain sizes for gravels or equal to bed form height in the case of sand-bed rivers [see Toro-Escobar et al., 1996], qsT is the (total) volumetric sediment transport rate per unit channel width, σ is the subsidence rate of the basement (assumed here steady at a given cross section but variable downstream), and λp is the porosity of the sediment deposit, taken to be constant. The independent variables are time t and the downriver spatial coordinate x.

[15] On long timescales, (∂η / ∂t + σ) >> ∂La / ∂t, that is, the active layer acts only as a transient reservoir that over geological timescales has negligible thickness variation. Thus, we neglect ∂La / ∂t and ∂(FL) / ∂t in equation (2). With these simplifications, and assuming quasi-steady sorting, equation (2) reduces to:

equation image

[16] Further, with the substitutions R = r / qsT (r = ∂η / ∂t + σ), introducing the scaling, x* = x / L and R* = (1 − λp)RL, where L is the length scale of the depositional alluvial basin [Paola and Seal, 1995], and by defining the relative mobility function as J = p / f, equation (3) takes the same form as given in the work of Paola and Seal [1995]:

equation image

[17] As pointed out in the work of Paola and Seal [1995], equation (4) is general and can be applied to describe any mass-conserving sorting process (i.e., selective deposition but not abrasion). The characteristics of the sorting mechanism are specified by the structure of the relative mobility function J, which could depend on downstream distance and the nature of the transport regime (i.e., bed load, suspension, mixed). R* is also a function of the distance x* and represents the relative distribution of sediment mass distribution by deposition down basin.

[18] A solution of equation (4) for the substrate grain-size distribution requires specification of the functions R* and J. Further simplification is achieved if in equation (4)R* is assumed independent of the downstream size variation and thus determined independently of solving equation (4). One obvious but restricted case where this is possible is depositional equilibrium, for which the deposition rate equals the subsidence rate [e.g., Paola, 2000]. A more general case is that for which the long-profile evolution is diffusional, with a transport (diffusion) coefficient independent of grain size. This is the case when the dimensionless shear stress is assumed constant [e.g., Paola, 2000; Marr et al., 2000] and means that the surface evolution is decoupled from the fining. This approximation holds only within either the gravel-bed or sand-bed regime, so we analyze these two cases separately.

2.1. Constant Dimensionless Stress (Shields Stress)

[19] The most important step in reducing the hydraulic aspects of downstream size sorting to a single similarity function is to assume constancy of the dimensionless shear stress, also known as the Shields stress, τ*, expressed customarily as:

equation image

where τo is the bottom shear stress, g is the gravitational acceleration, ρs and ρ are the sediment and fluid densities, respectively, and D is a characteristic sediment grain size. For bed-load-dominated rivers, which in the field usually equates to gravel-bed rivers, Parker [1978] provided the mechanistic basis by which bank erosion tends to widen the channel just enough to keep the shear stress slightly above critical. Since the grain roughness of such rivers is relatively high, the critical stress is constant when nondimensionalized. Field data supporting the constant-stress assumption for gravel-bed rivers can be found in the works of Parker [1978], Paola et al. [1992b], and Parker [2004], with typical values between 0.05 and 0.1. More surprisingly, Parker et al. [1998] investigated dimensionless stress variation in sand-bed rivers (bed-load-suspended load) and found that, although there was considerable scatter in the observed values, dimensionless stresses showed no systematic variation with slope or discharge [see also Parker, 2004]. For sands, the Shields stress values fall mostly between 1 and 2, more than an order of magnitude greater than the gravel-bed values. In parallel with this work, Dade and Friend [1998] proposed the following three constant-stress regimes: bed load (gravel bed), mixed load (sand bed), and pure suspension (silt). Church [2006] has proposed a continuum variation for the Shields parameter, based in transport modes and sediment characteristics. However, Church's plots show distinct clustering around the gravel and sand values given above. On the other hand, using the data set of Church and Rood [1983], Robinson and Slingerland [1998] have argued that, while some gravel reaches tend to maintain a constant value of the Shields stress, field data in general do not support the assumption of constant Shields stress.

[20] We summarize the debate as follows. There seems to be a good case for the constant-stress assumption for gravel-bed rivers, given both a strong mechanistic basis and support from observations. The sand-bed case is less clear. Additional data or more discerning analysis might reveal a trend in Shields stress that is not apparent now, and this prospect is hard to dismiss as long as there is no good mechanistic basis for maintaining constant Shields stress in sand-bed rivers. Nonetheless, we will proceed using the constant Shields stress assumption for both gravel-bed and sand-bed rivers, with different characteristic stress values for the two cases as discussed above. We admit the greater uncertainty for the sand-bed case, but in our view, even in this case, the constant-stress assumption is unlikely to introduce any more error into application of a sorting model to ancient sediments than would, for example, attempting to constrain the Shields stress from paleohydraulic reconstruction. And it has the distinct advantage of keeping the number of free parameters to a minimum.

[21] Bearing in mind that the dimensionless Shields stress should be considered as representative of a reach during a representative flood, the condition of constant averaged dimensionless stress will be used in the next section to derive summary relative mobility functions [J in equation (4)] from existing numerical models.

2.2. Distribution of Deposition

[22] Next, we turn to the downstream distribution of deposition R*(x*). As discussed earlier, we assume that the profile of deposition can be obtained independently of the downstream size profile, as long as we are within either the gravel-bed or sand-bed regime. Thus we assume that for a given problem, R*(x*) is known. An important point to recall here is the interesting invariant property of equation (4), previously pointed out by Paola and Seal [1995]: if a solution for the case R* = 1 of exponentially decreasing rate of deposition is known, then this solution is invariant for any spatial profile of deposition R*(x*) under the following transformation:

equation image

[23] Equation (6) defines a dimensionless distance transformation that accounts for spatial variation in the overall mass balance as deposition progresses downstream (i.e., y* is a deformed downstream coordinate that reflects the spatial distribution of R). Physically, it represents the idea that the faster sediment is removed spatially, the faster the size decreases, for a given input distribution and relative mobility function. If the solution for the reference case R* = 1 is, say, fR(x*), then the solution f(x*) for any other distribution R*(x*) is simply f(x*) = fR(y*), for a fixed J and input distribution. In this paper we focus on obtaining an analytical, similarity solution for the substrate fractions f for the general sorting case given by equation (4) and for the reference case R* = 1 [as explained, the same solution can be used for any distribution R*(x*) by virtue of the transformation (6)].

2.3. Self-Similar Solution

[24] As mentioned earlier, similarity approaches such as that given in equation (1) are commonly used for collapsing data. Similarity solutions have been used also when it is suspected that processes are dependent only on local scales and there is no other externally imposed temporal or spatial scale in the problem. According to Barenblatt [1996], self-similarity has a more fundamental significance beyond its elegance in collapsing many apparently disparate solutions into one; it also represents an intermediate asymptotic behavior for which the details of boundary/initial conditions do not influence the solution. That is, time and spatial scales remain sufficiently large in order to smooth out both fine-scale processes (for example, turbulence, small-scale details of flow and sediment transport dynamics) and initial/boundary condition influences but are at the same time small enough (thus “intermediate”) so long-term effects (for example, climate change, base level changes, etc.) do not influence the behavior of the system, dominated by its basic physics at these intermediate scales. Thus, the existence of such similarity solutions also provides insight into the influence of boundary conditions on system behavior. An obvious question then arises and is what would be appropriate intermediate time and spatial scales in fluvial systems, in particular when studying channel-driven sorting and downstream fining. In this paper we do not intend to discuss these concepts in depth, as they still remain much as open research questions in geomorphology. However, we anticipate that for the case of downstream fining presented herein, an intermediate timescales should in principle include many flooding (formative) events. This would represent an appropriate timescale where quasi-equilibrium, and thus conditions where our solutions are applicable, could be reached by the system.

[25] Here we propose a set of self-similar solutions motivated mainly by experimental observations of self-similar channel longitudinal profiles, final substrate grain-size distributions, and bed load composition for gravel mixtures [Parker et al, 1982; Snow and Slingerland, 1987; Paola et al., 1992b; Toro-Escobar et al., 1996; Seal et al., 1997]. Also, on the basis of the empirical observation of the approximate constancy of the cross-sectionally averaged Shields parameter [Parker et al., 1998; Dade and Friend, 1998], we propose that hydraulic controls on sorting (channel width, channel depth, and details of sorting mechanisms) can be collapsed into a regime-specific relative mobility function J, as appears in equation (4). Thus, the existence of a self-similar solution for f requires that both the substrate composition f and the relative mobility function J can be expressed as functions of the same similarity variable, say ξ, formed by local parameters defining the grain-size distribution. Given the generality of our treatment for J for the long term case, and assuming that the main sorting mechanisms can be captured in the relative mobility function J, then the primary control on the functional form of J and the form of the similarity variable ξ should be the transport mode: bed load, suspension, or a combination of these two. Transport regimes are in turn controlled by the dominant grain size (bed load for gravel-bed rivers, mixed bed load suspension for sand-bed rivers). The Shields stress τ* is assumed constant within either regime.

2.4. Derivation of the Relative Mobility Function

[26] At this point we wish to apply our constant-Shields-stress assumption to determine general forms for the relative mobility function J. We propose that the form of ξ differs according to the value of τ*, with two characteristic limiting cases representing unimodal distributions for either gravels or sands. In dimensional units, both mean and standard deviation of surface and subsurface size distributions decay downstream at comparable rates for gravels [e.g., Paola and Seal, 1995; Paola and Wilcock, 1992], suggesting that the coefficient of variation (standard deviation/mean) is approximately constant. For sand-bed rivers, field observations indicate that standard deviations decay at much slower rates than mean sizes [White et al., 1978], that is, the standard deviation is approximately constant. On the basis of these results, we propose the following forms for the similarity variable ξ:

[27] 1. When τ* ≈ 1.4τ*c (gravels):

equation image

[28] 2. For τ* ≈ 1 − 2 (sands)

equation image

where D is a given sediment size, and equation image(x*) and σ(x*) are the local mean and standard deviation of the mixture in transport, respectively (all these are dimensional quantities with units of length). The scaling represented in equations (7) and (8) can be collapsed into a single compact form:

equation image

where Cv is the coefficient of variation of the material in transport and the exponent β takes the value 0 for the case of gravels and 1 for sands. Because we seek a general similarity solution to equation (4) for the case R* = 1, the dependence on the spatial coordinate x* (or on the transformed depositional length y* for the general case) is parametric only, via equation image(x*) and σ(x*) through the definition of ξ. These last two parameters are the linkage between the surface dynamics and the deposited grain size.

[29] Given these two general forms for the similarity parameter ξ, equations (7) and (8), we now use results from existing semi-empirical, hydraulically based fining models to determine the relative mobility function J. The models have been previously calibrated and tested against field and experimental data. We ran the models under steady state subsidence and sediment supply over long periods of time (5000 to 10,000 years), with arbitrary values of input water and sediment discharge and input grain-size distributions for the sediment feed. In all cases tested here (a variety of initial and boundary conditions) for both sand or gravel-bed rivers, constant water and sediment discharge were used in a given run, together with constant composition (size distribution) of the sediment feed and constant river width throughout the total basin length. The initial composition of the substrate was always taken equal to the input value for the material in transport at the feed point upstream, but it evolved throughout a given simulation. The models were allowed to run and evolve to steady state as the fluvial system developed toward a base level that was kept constant at all times. No specific value of the dimensionless Shields parameter was set prior to calculations, as the models compute the Shields stress internally. Instead, other model variables (for example, width, discharge) were manipulated to produce the target Shields stress, on the basis of the values presented above.

[30] The model ACRONYM [Parker, 1991] was used for gravel-bed rivers, whereas a model developed by Wright [2003] was chosen for the case of low-gradient sandy-bed rivers. In the case of gravels, ACRONYM includes expressions for fractional bed load transport that have self-similar forms; in this case, these similarity expressions are largely based on empirical observations [Parker, 1991]. On the other hand, the model used for sand-bed rivers includes more complex fractional transport expressions that are not self-similar [see Wright, 2003]. It is thus especially interesting that similarity emerges spontaneously in the sandy river numerical model when run for long time periods.

[31] Using these hydraulic models [Parker, 1991; Wright, 2003], our first step was to compute the relative mobility function J = p / f and verify the validity of the similarity assumption and scaling proposed in equations (7) and (8), as the numerical models allow for the calculation of fractions in transport p and fractions in the substrate f of a given sediment size, at different downstream positions. Figures 1 and 2show the relative mobility function J = p / f, as a function of our similarity variable ξ, computed with these models, for the case of gravel-bed rivers (ACRONYM, Parker [1991]) and sandy low-gradient rivers [Wright, 2003], respectively. Both Figures 1 and 2 show plots of computed relative mobility functions at different locations downstream of fluvial systems with exponentially decreasing rates of deposition (i.e., the reference case), after 5000 years, using our similarity variable (9) with the corresponding β value as discussed above. Figures 1 and 2 also include a best fit for the relative mobility function in each case. For gravels, the best fit has the general form:

equation image

with ag = 0.9, bg = 0.2, and cg = 0.15 for the case shown in Figure 1. For sands, the best fit has the following general form:

equation image

with as = 0.8, bs = 2.5, and cs = 0.15, for the case shown in Figure 2. We also note here that, as seen in Figures 1 and 2, our similarity variables do not fully collapse the numerical results at the fine end of the gravel range and the coarse end of the sand range.

Figure 1.

Self-similar relative mobility function for gravel-bed rivers with bed load. Numerical model results from ACRONYM [Parker, 1991] (symbols), and fit given by equation (10) (line).

Figure 2.

Self-similar mobility function for sand-bed rivers with suspension. Numerical model results using a formulation developed by Wright [2003] (symbols), and best fit, equation (11) (line).

[32] The approximately self-similar forms for the relative mobility functions in both gravel and sand-bed rivers shown in Figures 1 and 2 support the idea that the averaged behavior of the hydraulic and grain-interaction processes that account for the details of differential mobility and transfer of sediment sizes from transport to permanent storage in the substrate can be collapsed, via the definition of our relative mobility function J. This new relative mobility function J is, in turn, expressed in self-similar forms that depend only on the local characteristics of the sediment distribution in transport (mean and variance). Thus, using the generalized functions J = J(ξ) for each of the limiting cases (gravel or sand), we now proceed to find a solution f = f(ξ) of equation (4) for the reference case R* = 1.

[33] For gravels (β = 0), substitution of J = J(ξ) and f = f(ξ), and the use of the chain rule lead to:

equation image

whereas for sands (β = 1), we find:

equation image

where f′ and J′ indicate the derivatives with respect to the similarity variable ξ. Now if the functions f and J are to be self-similar, then all the coefficients of f′ / f, ξf′ / f, J′ / J and ξJ′ / J in equations (12) and (13) cannot depend explicitly on x* (self-similar solutions require that dependence on the spatial variable be parametric, via the similarity variable ξ only). Thus, for the case of gravels we have:

equation image
equation image

and the governing equation (12) becomes:

equation image

For sands, we find the following:

equation image

since we have taken the standard deviation to be constant. The governing equation (13) for sands becomes:

equation image

The constants C1, C2, and C3 are so far undetermined and must be found empirically. We can, however, advance an interpretation of the role of these constants on the basis of physical reasoning. C1 and C2 express the relative partitioning of variance in the gravel supply between downstream change in mean size (C2) versus standard deviation (C1). On the other hand, C3 expresses the fraction of the total variance of supply sand size that is partitioned into downstream fining of the mean size, given that we hold the local standard deviation constant (for instance, if deposit standard deviation were everywhere equal to the supply standard deviation, then there would be no fining of the mean sand size and C3 would be 0). We provide computed ranges for these constants below.

[34] Solving equations (14a) and (14b) for gravel leads to:

equation image
equation image

and for sands, to simply:

equation image

[35] In equations (18a), (18b), and (19), equation image, σ0 are the mean and standard deviation of the input sediment-size distribution (dimensional), respectively, for each corresponding case (gravel or sand). It is noteworthy that both equations (18b) and (19) have the form of the well-known Sternberg formula [Sternberg, 1875], but were derived for fining by selective deposition rather than abrasion. In particular, equation (19) (sand) shows the physical meaning of the effect of the overall sedimentation pattern on fining, when compared to the usual Stenberg expression D = Doeax (here D is sediment size and the index o indicates the initial value at the upstream end of the depositional system). Equation (19) is given in terms of a dimensionless distance x* = x / L(t) that scales the overall rate of sediment extraction by deposition down the basin. Thus it embodies the simple effect of elongation of equilibrium fining profiles of two different size basins with similar size ranges in their feed points and distal ends [Paola et al., 1992a; Hoey and Ferguson, 1997; Robinson and Slingerland, 1998; Hoey and Bluck, 1999; Paola, 2000]. Generalizing the definition of the decay coefficient a in the Stenberg formula to a = a′ / L would explicitly account for this length-scale effect and would at least partially explain the wide range of values of a determined by many authors [see Rice, 1999].

[36] Further, inspection of equations (18a) and (18b) [or alternatively, equations (14a) and (14b)] indicates that for the case of gravels, the mean and standard deviation decay at approximately the same rate, and so, in that case, the coefficient of variation Cv must remain approximately constant, whereas for sands, the standard deviation has dropped out of the analysis and therefore must remain constant. As a consequence, the coefficient of variation for sands should increase downstream at a rate C3, similar to that of the decay of the mean, according to these results. In effect, the ratio between conditions (14a) and (14b) leads to:

equation image


equation image

provided C1 and C2 remain approximately constant. This indicates in turn that, for gravels, the coefficient of variation of the distribution of sizes in transport should remain approximately constant. The case of sands is different; since we assumed that the local standard deviation of material in transport does not change appreciably downstream, we then have:

equation image

which indicates that the coefficient of variation of the material in transport for sand-bed rivers should increase downstream.

[37] Figure 3 shows the computed coefficients of variation for material in transport for the cases of gravel and sands using the full hydraulic numerical models [Parker, 1991; Wright, 2003]. These results together with the field data presented in Table 1 support the results implicit in equations (21) and (22) above. Interestingly, experimental observations and numerical calculations suggest that downstream changes of the substrate coefficient of variation follow similar trends as those corresponding to the material in transport, for both gravels and sands, as shown in Figure 4. In addition, Figure 5 compares field and experimental data that show an approximately constant coefficient of variation of the bed surface material for gravels as well.

Figure 3.

Computed coefficient of variation for the material in transport for the cases of gravel and sand (results from numerical simulations).

Figure 4.

Substrate coefficient of variation for gravels and sands. Numerical simulations, laboratory experiments, and field data.

Figure 5.

Coefficient of variation of the bed surface material for gravel-bed streams (field and laboratory data).

Table 1. Field Observed Values of the Coefficient of Variation Cv of Bed Surface Material (Computed as the Ratio Between Standard Deviation and Mean Size in Meters) for Gravels and Sands [White et al., 1978; Ferguson et al., 1996]a
Gravel-Bed StreamsSand-Bed Streams
  • a

    In the case of gravel-bed river reported in the work of Ferguson et al.[1996], the values used here were obtained from the size-cumulative curves of bed surface material corresponding only to gravels with negligible material smaller than 2 mm.

Aare (1)0.87Mountain Creek3.5
Aare (2)0.78Niobrara4.25
Elbow River0.74Mississippi at St. Louis3.23
Ferguson et al. [1996]0.7–0.8Mississippi-Tarbert (LA)2.25

[38] The governing equations (15) (gravel) or (17) (sand) can be readily solved, given that the mobility function, J = J(ξ) for each case is known [i.e., equations (10) and (11)]. The general solution of equation (15) or (17) is:

equation image

where for the case of gravels, the integrand takes the following form:

equation image

and for sand:

equation image

The integration constant C that appears in the general solution of the problem, equation (23), can be evaluated from the mass-conservation condition:

equation image

[39] Figures 6 and 7compare the analytical solution with numerical results for the cases of gravel and sand, respectively. The solution presented above, equation (23), is valid as long as both R* > 0 and J(ξ) ≠ constant (that is, there must be deposition and selective transport in order to develop downstream sediment sorting), and for unimodal distributions, with the corresponding definitions of the similarity variable given above for the gravel and sand cases.

Figure 6.

Approximately self-similar substrate grain-size distributions for the case of gravel rivers. Numerical results (ACRONYM; Parker [1991], dashed lines) for different downstream locations, and analytical solution (full line).

Figure 7.

Approximately self-similar substrate grain-size distributions for the case of sand-bed rivers. Numerical results [Wright, 2003] (dashed lines) for different downstream locations, and analytical solution (full line).

2.5. Values of the Undetermined Constants

[40] Although we do not claim universality of the constants C1, C2, and C3, it is likely that their numerical values are restricted. To test this idea we performed numerical calculations for the reference case using a range of values for input parameters (water and sediment discharge and input size distribution). This numerical experiment suggests restricted ranges for both C1 and C3, 0.5 < C1 < 0.9, with an average value for C1 ≈ 0.75 for gravels, and 0.1 < C3 < 0.45, with an average value of C3 ≈ 0.3 for sands.

2.6. Gravel-Sand Mixtures

[41] So far, we have assumed that the bed sediment is either gravel or sand. We also investigated the case of 0 < β < 1, which might be useful for intermediate values of dimensionless stress and/or for bimodal distributions with a single equation. Using the definition of the similarity variable [equation (9)] and following a similar analysis to that above for the limiting cases of gravel or sand, three analogous conditions to those given in equations (14a), (14b), and (16) for the existence of self-similar solutions can be found. However, independent solutions for these conditions do not appear to exist for any value of β ≠ 0 or 1. Therefore we conclude that for this intermediate case, self-similar forms do not exist and the solution for the substrate grain-size distribution f must depend explicitly on the spatial variable x* (or its transformed y*).

2.7. Fining and the Long Profile

[42] Although we have not focused here on the related problem of self-similarity of the long river profile, one of the motivations of our work is the experimental observation of approximately self-preserving grain-size distributions linked to self-preserving long topographic profiles [see Toro-Escobar et al., 1996]. For the reference case defined above, the downstream variation of sediment transport rate is of the form eλx* and produces also approximate self-similar channel long profiles of the form given in equation (1) above [see also Toro-Escobar et al., 1996]. In eλx*, λ is the downstream rate of decay in sediment transport (presumably taking different values for gravels or sands), and x* = x / L(t) is the dimensionless depositional distance [note that this includes the scaling factor L(t) for self-similar long profiles given in equation (1)]. Now assuming that qsTequation imageτ*m, and if also equation imageeC3x*, [see equation (19)], combining all these, we then obtain C3 ≅ 2/3λ (for the reference case). The same result can be arrived to for C1 (gravel), while the value of C2 can then be obtained using equation (20) and the initial conditions. Therefore this last relation C3 ≅ 2/3λ is the link between channel long profile and fining, since a solution of for the long profile ∂η / ∂tλeλx* must include both λ and the scaling parameter L(t).

3. Application of the Self-Similar Solutions

[43] Our goal in this paper has been to show that, on large time and spatial scales, most of the variation in sorting patterns in river systems is controlled by the depositional mass balance (because fining is driven by preferential extraction of less mobile sizes) and the supplied size distribution (because the supply sets the range of sizes on which selective transport can act). Our relative mobility function, which summarizes the hydraulics and particle dynamics, acts on the normalized input size distribution at a given river cross section to determine the range of sizes that will be extracted to permanent storage, thereby setting the overall downstream fining. The downstream profile of deposition then determines the arrangement of median sizes in space. This summarizes the general idea behind our approach. Now we provide a general description of how to implement these results in stratigraphic models.

[44] As mentioned before, we have assumed that for given initial and boundary conditions (total sediment and water discharges, input sediment mixture, tectonic subsidence pattern, and base level variation), the deposition profile (i.e., the distribution R*(x*)) is known over the time interval of interest. Procedures to calculate topographic evolution and deposition profiles have been extensively described elsewhere [e.g., Flemings and Jordan, 1989; Slingerland et al., 1994; Paola, 2000; Marr et al., 2000]. In addition, if both gravels and sands are input to the system, we assume that each will constitute a separate regime [see Marr et al., 2000], within which, the model presented here can be applied. In that case, the gravel-to-sand transition must be determined as an internal moving boundary [Paola et al., 1992a; Sambrook-Smith and Ferguson, 1995; Parker and Cui, 1998; Marr et al., 2000; Swenson, 2000], since it determines the total depositional length L for each regime. Then the downstream variation of the parameters describing the mixture in transport can be computed using equations (18a) and (18b) for gravels, or equation (19) for sand, together with equation (6) for any distribution of deposition R*(x*), within each regime. With the forms for the relative mobility function given by equation (10) (gravel) or (11) (sand), and the similarity variable known at each location, the size distribution of the substrate can easily be computed using the general solution, equation (23), for each particular case (i.e., gravel or sand).

4. Testing the Analytical Model Results

[45] Directly testing our model predictions requires data on (1) downstream variation of deposition rates and (2) cross-sectional average substrate composition over enough of the depositional system to allow reconstruction of the input size distribution. As minimal as these requirements seem to be, it is surprisingly hard to find data sets satisfying them together. Here we test our analytical model using field and laboratory data including both gravel and sand-bed streams. For the gravel case, we use a field data set reported by Seal and Paola [1995] collected in the North Fork Toutle River (Washington State) and laboratory observations reported in the work of Seal et al. [1997]. For sand-bed rivers, we test our predictions with the same data set used in the work of Wright [2003], including both the lower Mississippi River and the Fly River in Papua, New Guinea, verifying in this way the predictive capabilities of the simplified analytical model presented here.

[46] The field data set of the North Fork Toutle River [Seal and Paola, 1995] includes a large number of measurements of bulk (substrate) size distributions at different cross sections, together with the deposition profile produced after several years. Here we use this data set to compare observed and computed values of downstream changes in both the bulk (substrate) mean size (Figure 8a) and standard deviation (Figure 8b), along with the observed and theoretical substrate grain-size distribution [solution of equation (23)], as a function of our similarity variable (Figure 9). Although the data reported in the work of Seal and Paola [1995] indicate that the North Fork Toutle River transports both gravels and sands, both field experience and measured data suggest that (1) bed load is the dominant mode of transport and (2) measured deposited size distributions are gravel dominated. Therefore, in our calculations (Figures 8 and 9), we treat the entire North Fork Toutle system as gravel-dominated. For this case, both known deposition rates and the geometry of the deposit were used to infer downstream values of R* [see Strong et al., 2005], which in turn allowed us to calculate of downstream changes in the variable y* that is used in expressions (18a) and (18b) for the mean and standard deviation, respectively (Figure 8). In addition, we also used (1) the similarity variable for gravels given by equation (7); (2) an average value of the constant C1 = 0.75 [the constant C2 is obtained from equation (22) and the initial input mixture]; (3) equation (10) for the relative mobility function (bed load dominated and gravels) with ag = 0.9, bg = 0.2, and cg = 0.15; (4) a constant ratio between mean size of substrate and material in transport of 2.5 (based on the similarity solution for gravel shown in Figure 6); and (5) the initial input grain-size distribution was obtained after integrating the composition of the entire deposit [Paola, 2000], using measured substrate grain-size distributions and geometry of the deposit [Seal and Paola, 1995]. Figure 8 shows good agreement between observed and predicted total amount of fining although the theoretical prediction does not follow the observed downstream fining profile exactly. Figure 9 also shows general agreement between observed and theoretical substrate size distributions. We speculate that the error in both Figures 8 and 9 is mainly a consequence of the presence of mixed sizes (gravel and sand).

Figure 8.

Observed [Seal and Paola, 1995] and computed downstream changes in (a) substrate mean size and (b) substrate standard deviation, for the North Fork Toutle River (gravel).

Figure 9.

Theoretical (solid line) and measured [Seal and Paola, 1995] (symbols correspond to different dimensionless downstream cross sections) substrate grain-size distributions for the North Fork Toutle River.

[47] A procedure analogous to that explained above for the case of the North Fork Toutle River was used to compare model predictions and laboratory data on gravel-bed experiments reported in the work of Seal et al. [1997] (including also C1 = 0.75). Predicted and observed downstream trends of deposit mean size and coefficient of variation Cv are given in Figures 10a and 10b, respectively. Finally, the sand-bed case was tested using known bed long profiles and measured characteristic mean size for both the lower Mississippi River and the Fly River (New Guinea) [Wright, 2003] (Figures 11 and 12, respectively). Known bed profile concavity was used to infer the downstream changes in R* and thus y*, and a value of C3 = 0.1 was used in both cases. In the case of the lower Mississippi River, initial conditions were reset at 900 km downstream Cairo, IL (see Figure 11), since the long profile shows a change in concavity, presumably as a consequence of input from a major tributary, the Arkansas River. Results are satisfactory for both the lower Mississippi (Figure 11) and the Fly River (Figure 12). To finalize, we mention that although rivers such as the Mississippi or the Fly River might appear, at first glance, to be far from any equilibrium (the Mississippi River has been the subject of repeated engineering interventions in the last century; the Fly River is subject to increased sediment inputs because of mining activities), our proposed self-similar solutions and assumptions appear to be valid, as model results approximately duplicate observed trends for a wide range of spatial and temporal scales (see Figures 8, 9, 10, 11, and 12).

Figure 10.

Observed and computed downstream changes in (a) substrate mean size and (b) substrate coefficient of variation, for the laboratory experiments of Seal et al. [1997] (gravel, narrow channel).

Figure 11.

Observed and predicted downstream changes in mean size for the lower Mississippi River (initial conditions were reset at x = 900 km downstream Cairo, IL, at the Arkansas River confluence).

Figure 12.

Observed and computed downstream changes in mean size for the Fly River, Papua New Guinea.

5. Concluding Comments

[48] A simplified self-similar solution for the long-term substrate grain-size distribution, based on a self-similar form for a relative mobility function J, can collapse the fine scale details of the hydraulics and particle dynamics of fluid sediment sorting. In the model proposed here, the local selectivity, i.e., the range of sizes that can be deposited together at one location given a set of hydraulic processes, is set by the relative mobility function (J).

[49] The relative mobility functions we propose differ for gravel and sand, reflecting differences in fractionation mechanisms for bed-load-dominated versus mixed-suspension and bed load systems.

[50] We present two different forms of the similarity variable for the limiting cases of sand or gravel-bed rivers. This scaling allows for a variable coefficient of variation of material in transport in the cases of sand-bed rivers, whereas the coefficient of variation of transported mixtures is approximately constant in the case of gravel-bed rivers. Deposited gravel size distributions conserve the coefficient of variation while sand-size distributions tend to decrease in mean size without becoming relatively better sorted.

[51] Overall, the similarity approach we present suggests that most of the variation in observed downstream fining profiles in fluvial systems is the result of the following two fundamental controls: input size distribution and downstream distribution of deposition. In sedimentary basins, the latter can potentially be constrained through chronostratigraphic correlation. If the former can be constrained from, for example, well data, the approach presented here would provide a means of interpolation or extrapolation of grain size in the subsurface.

[52] The model can be applied to investigate the long-term steady state of basins subject to different patterns of tectonic subsidence and can be easily incorporated into more complex stratigraphic numerical models without requiring simulation of all fine-scale hydraulic and sediment transport processes.


[53] We thank the Office of Naval Research for support through its EuroSTRATAFORM Program, grant N00014-04-1-0556. This work was also supported by the National Center for Earth-surface Dynamics, NSF EAR-0120914. We also acknowledge the constructive suggestions of M. Church and R. Ferguson and two other anonymous reviewers who helped improving a manuscript of this paper.