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Corresponding author: W. B. Dade, Department of Earth Sciences, Dartmouth College, Hanover, NH 03755, USA. (william.b.dade@dartmouth.edu)

Abstract

[1] The ability to understand and predict the flux and fate of sediment delivered to the sea by rivers remains an outstanding scientific challenge. Approaches to this challenge are necessarily synthetic, spanning wide ranges in spatial and temporal scales. Here a conventional sediment transport theory used by engineers and sedimentologists at reach and channel scales is applied at the basin scale. Specifically, a straightforward expression proposed by Bagnold and modified accordingly predicts the observed importance of combined wetness and steepness of a source basin as a control of sediment supply to the sea. The reasonable, key assumption underlying the application of sediment transport theory in this context is that the river-mouth sites for which suspended-sediment loads are reported are alluviated, and thus characterized by transport-limited flux of sediment. This analysis also indicates the potential significance of additional, as yet poorly documented factors constraining sediment supply to the sea. These factors, some of which appear to covary systematically with climate, include river-profile concavity, river-mouth channel width and friction, and the characteristic size of sediment in transport.

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[2] Each year approximately 20 billion tons of solid detritus and dissolved materials are shed by the world's land masses into the coastal ocean [Holeman, 1968; Milliman and Meade, 1983]. These fluxes to the sea, delivered primarily by rivers, are measures of the rates at which continents wear down in response to tectonic uplift, affect nearshore marine environmental quality and geomorphology, and provide the raw materials of the marine siliciclastic sedimentary record. Accordingly, the challenge of understanding and predicting particulate sediment flux to the sea has received attention from geomorphologists, geochemists, geophysicists, and marine sedimentologists alike [Ahnert, 1970; Pinet and Souriau, 1988; Milliman and Meade, 1983; Milliman and Syvitski, 1992; Summerfield and Hulton, 1994; Mulder and Syvitski, 1996; Hovius, 1998; Harrison, 2000; Milliman and Farnsworth, 2011]. These studies have typically taken a formal statistical approach or a graphical, exploratory approach. Each has highlighted various empirical relationships between sediment supply and basin relief or precipitation, or both. As the number and range of observations have increased, so too has the secondary importance of basin size, regionally averaged seasonal temperatures, as well as underlying geology been recognized [Milliman and Syvitski, 1992; Harrison, 2000; Milliman and Farnsworth, 2011].

[3] One potential drawback of these existing approaches is that relationships have been considered only among relatively readily known, basin-scale quantities, and do not necessarily uncover what details need to be known better to improve our understanding. Here a physics-based formula for sediment transport is applied to the most comprehensive set of data for sediment discharge by rivers to the sea yet reported [Milliman and Farnsworth, 2011]. New insights emerge regarding details that warrant further study.

2. Analysis

[4] The basis of any deterministic approach to natural phenomena is the assumption that there exists a direct, physically significant correspondence between driving, independent variables and the dependent variable(s) of interest. In predicting sediment transport in a river, conventional, independent variables are water discharge, channel geometry and slope, bed shear stress, and the characteristic size and density of individual particles in transport. Derivations of deterministic formulae for suspended-sediment flux typically reflect a balance between, say, the energy required to maintain the buoyancy flux associated with the suspended load and the available turbulent kinetic energy in the water column. For one such derivation see, for example,Dade and Friend [1998]. The details of individual sediment transport functions can be as varied as the practitioners who generate them. Summaries and comparisons of such formulae can be found in the works of Soulsby [1997], Yang [2006], and Julien [2010], among other sources.

[5] In deterministic transport functions, sediment flux is typically related to the Shields parameter θ, which is a dimensionless ratio of sediment-transporting shear force and submerged particle weight; thus,

θ=τg(ρs−ρ)d,

where τ is bed shear stress, ρ_{s} and ρare individual-particle and fluid densities, respectively,d is characteristic grain size in transport, and gis acceleration due to gravity. Bed shear stress in a river channel can be evaluated using the depth-slope product and, upon introduction of a friction coefficientf ≡ τ/ρu^{2}to eliminate flow depth in terms of average water-flow speedu. Thus, in turn, in terms of water discharge, the Shields number can be given equivalently as

θ=(fQ2s2gR3w2d3)1/3.

In equation (2), Q is volumetric discharge of water (L^{3}T^{−1}), s is local channel slope, w is local characteristic channel width, and R≡(ρs–ρ)/ρ is relative excess density of individual particles in transport.

[6] Deterministic formulae for significant volumetric flux q_{s} (L^{2}T^{−1}) of fine, noncohesive sediment per channel width can be expressed in the form

qsν=α0θa,

where ν is kinematic viscosity of water, α_{0} is a dimensionless parameter that may be a function of channel friction f and grain size d depending on the transport formula considered, and the exponent a takes on values in the range {2, 8} depending again on the specific formula as well as the range of grain sizes considered. Upon substitution of equation (2), equation (3) can be given equivalently as

qsν=α0(fQ2s2gR3w2d3)a/3.

Upon multiplying both sides of equation (4) by channel width w and viscosity ν, one obtains an expression for volumetric sediment flux Q_{s} (L^{3}T^{−1}) given as

Qs=α0νw(fQ2s2gR3w2d3)a/3.

Equation (5)indicates that the volumetric rate of sediment discharge to the sea from any given river, to the degree that it is transport-limited, is a function of river-mouth water discharge, channel width, slope and friction, and the caliber of material in transport. The assumption of transport-limited control of noncohesive sediment flux is a reasonable one for many if not most river-mouth sites, which are typically alluvial or at least partially alluviated with loose sediment. The assumption would not necessarily apply, however, to exclusively bedrock-bound catchments, or to river-mouth channels and adjacent floodplains comprising strongly cohesive sediment.

[7] Given the potential generality of equation (5)and the problem at hand, it is useful to restate this expression in terms of basin-scale properties. Characteristic river-mouth discharge of waterQ, for example, reflects the product of source basin area Aand basin-average effective precipitationP (that is, runoff, or precipitation in excess of losses to evapotranspiration and infiltration).

[8] Local channel properties such as channel width, however, may or may not be related to basin-scale characteristics. The caliber of sediment delivered to the sea may, too, be as variable as the underlying bedrock geology, geologic history, size, and climate of individual basins. It is documented, for example, that the median grain size of fluvial-suspended sediment can easily span a range of two orders of magnitude, from <2 μm in tropical and subtropical rivers in terrain with deeply weathered soils, to 150 μm in large, low-lying rivers draining the once-glaciated Siberian tundra [Walling and Moorehead, 1989].

[9] Sediment discharge can be related to average steepness of a basin, at least, if rivers exhibit self-similar longitudinal profiles, such that local channel slopes(x) at any distance x downstream from the headwaters be adequately described as

s(x)=s0(x0x)1−b,

where s_{0} and x_{0} are arbitrary reference values. The value of b introduced in equation (6)is a measure of longitudinal-profile concavity, and is less than 1 for river profiles that are typically concave upward; otherwise variability inbfrom river to river can reflect basin-specific spatial patterns of uplift and subsidence, tributaries adding water and sediment to the main channel, and comminution of grains in transport, among other things [Shepard, 1985; Snow and Slingerland, 1987; Sinha and Parker, 1996]. If equation (6) adequately applies in this instance, however, average steepness of a river basin S ≡ H/L, where H and Lare overall basin relief and length, respectively, is related to local river-mouth slopes(L) = bS. Upon substitution of this result, equation (5) becomes

Qs=α0νw(fb2Q2S2gR3w2d3)a/3.

Upon normalization of each independent parameter by its global average for dimensional consistency, such that for any independent parameter Y,Y*≡Y/Yavg, and identifying appropriate reference values Z_{0} such that for any dependent variable Z,Z*≡Z/Z0, equation (7) can be expressed as

Qs*=α1(Q*S*)2a/3,

where α1=α0*b*2a/3f*a/3w*1−2a/3d*−a. Re-expressingequation (7) in this way enables straightforward tests of the physical significance of the interaction of wetness and steepness, and of the functional form of a particular transport formula of interest. Moreover, variability in sediment flux unexplained by equation (8) can be attributed in part to factors rendering α_{1}≠ 1; that is, to variability in channel profile-concavityb, width, w, friction f, and grain size in transport d.

[10] Of interest here for its simplicity and effectiveness is the case for which a = 2 and α0∝f−1; this corresponds to Bagnold's expression for suspension transport [Bagnold, 1966; Dade and Friend, 1998]. In that particular case

Qs*=α2(Q*S*)4/3,

where α2=b*4/3f*−1/3w*−1/3d*−2. Equation (9) indicates that volumetric flux of sediment to the sea Q_{s} is proportional to the product of volumetric discharge of water Q and average steepness of a basin S raised to the ^{4}/_{3} power, all else being equal. The quantity ρgQS is a measure of average stream power per length of river or, equivalently in this case, the rate at which the work of transporting suspended sediment to the sea can be achieved by any given river.

[11] Geomorphologists are often as interested in sediment yield, or, equivalently, volume of particulate detritus removed per area of source river basin. This quantity is equivalent to the basin-wide average erosion rateE (mm yr^{−1}) at which a landscape wears down due to weathering and, in some settings, in response to active tectonic uplift of topography and/or rock. Dividing both sides of equation (8) by basin area A and again normalizing by globally averaged values and reference values yields the general expression

E*=α1(P*S*)2a/3A*2a/3−1,

and, in the specific case of Bagnold suspension transport,

E*=α2(P*S*A*1/4)4/3.

[12] Finally, in the context of this analysis, values of the ratio α2=Qs*/(Q*S*)4/3=E*/(P*S*A*1/4)4/3for individual river basins provide basin-specific constraints on channel concavity and grain size in transport, among other factors. Insection 3 the relevance of this analysis is explored in the context of data pertaining to more than 400 rivers worldwide compiled and reported by Milliman and Farnsworth [2011].

3. Wetness, Steepness, and Regional Climate

[13]Milliman and Farnsworth [2011]have compiled the world's largest database for river discharge of water and suspended sediment to the coastal ocean, as well as source-river basin area, length, relief, and climate. The 1534 rivers listed in this compendium drain more than 85% of the Earth's land surface. All values of sediment load delivered to the sea reported byMilliman and Farnsworth [2011]pertain to river-mouth locales. As mentioned earlier, such settings are typically alluvial or at least partially alluviated with loose sediment, and so amenable to the analysis pursued here.

[14]Milliman and Farnsworth [2011] candidly acknowledge potential problems with the data. These include, briefly, (1) measurements and inferred values of questionable quality and/or comparability, (2) inadequate duration of river monitoring even if of high quality, (3) incomplete documentation of regulated (dammed or diverted) rivers, and (4) human error in the recording and transcription of data. Milliman and Farnsworth [2011] cite, as an example of issue 1, that the range of six previously reported estimates of the straightforward measure of basin area A of the Amazon River alone corresponds to ∼20% of their own estimate of 6.3 × 10^{6} km^{3}. Thus, to the list above can be added (5) a lack of precision owing to the compiling authors' reasonable practice of reporting values rounded to only a few significant figures.

[15] Of the more than 1500 cases reported by Milliman and Farnsworth [2011], 735 include data for sediment discharge Q_{s}_{.}Of these, 38 cases are excluded here for being heavily regulated rivers with values of the ratio of documented discharges for pre- and postdam erasQ_{post}/Q_{pre}<0.5. Of the remaining cases, 402 include the values for basin area, length, and relief required to be considered in this analysis. These 402 rivers collectively drain about one-third of the Earth's land area that overall sheds on average just under two km^{3}, or equivalently, about 5 billion metric tons, of sediment each year.

[16] Listed in Table 1 are global averages, for the 402 rivers considered here, of basin area A, steepness S, water discharge Q, and effective precipitation (or runoff) P; also listed in Table 1 are reference values of volumetric sediment discharge to the sea Q_{s} and basin denudation rate E for use in equations (9) and (11). These quantities are used to normalize values in Figures 1 and 2, as indicated by the subscript (_{*}) in each figure.

Table 1. Global Averages of Basin Size, Slope, Water Discharge, and Runoff^{a}

A_{avg} (10^{3} km^{2})

S_{avg}

Q_{avg} (km^{3} a^{−1})

P_{avg} (mm a^{−1})

Q_{s}_{0} (km^{3} a^{−1})

E_{0} (mm a^{−1})

a

Reference values are for sediment discharge and denudation rate.

122

0.01

49.3

719

0.02

0.48

[17] Shown in Figure 1 are values of normalized volumetric sediment flux to the sea Q_{s} as a function of normalized stream power QS. The solid line in Figure 1 indicates the relationship given in equation (9) for α_{2} = 1, which accommodates ∼85% of the variability in Q_{s}.

[18] Shown in Figure 2 are normalized values of basin erosion rate Eas a function of the normalized wetness-steepness parameterPSA^{1/4}. The solid line in Figure 2 indicates the relationship given in equation (11) for α_{2} = 1, which accommodates ∼25% of the variability in E.

[19] Shown in Figure 3 are the median and range of the 25th and 75th quartiles of α_{2} and, for comparison, the ratio εof mechanical-to-chemical weathering of individual rivers as functions of regional climate as defined and reported byMilliman and Farnsworth [2011]. Climatic categories are delineated by Milliman and Farnsworth as follows:arctic: summer temperatures >0°C, winter temperatures ≪0°C; subarctic: summer >10°C, winter >0°C; temperate: summer >10°C, winter >0°C; subtropical: summer >20°C, winter >10°C; tropical: summer and winter temperatures >20°C.

4. Discussion and Conclusions

[20] An inspection of Figure 1indicates that conventional sediment transport theory provides useful constraints for the volumetric flux of suspended sediment to the sea. This analysis focuses on but one possible, physics-based formula for fine-sediment transport, that because ofBagnold [1966], chosen for its simplicity and effectiveness. Other, more complicated transport functions could be considered, requiring more computational effort for marginal return, and certainly not making the analysis any more accessible. The key point here is that, upon applying this particular formula to basin-scale properties, the rate of sediment supply is predicted and observed to a significant degree to reflect the interaction of wetness (reflected in average annual dischargeQ) and steepness (reflected in basin-averaged slopeS), and hence the average stream power per length of a river.

[21] Related predictions of basin-averaged erosion rateE from equation (11) and shown in Figure 2suggest that a deterministic formula for sediment transport can yield the functional form of the relationship between erosion rate and basin steepness, wetness, and size, but predictions compare with observations in detail with only mixed success. The reduction in explanatory power of sediment transport theory follows in part from the elimination of the effects of basin size. That is, consideration of sediment load as some function of water discharge at the basin scale necessarily requires the consideration of the product of sediment yield and area as a function of the product of basin-averaged precipitation and basin area; good agreement simply reflects, to some degree, the effect of basin size. In the context of this analysis, however, remaining, unexplained variability in basin area-specificEcan be attributed in part to undocumented variability in river-profile concavity, river-mouth channel width and friction, and the characteristic size of sediment in suspension transport. These factors are embodied in the coefficientα_{2.} As indicated in Figure 3, values of α_{2}exhibit wide-ranging variability that may nevertheless covary systematically with climate. Systematic variability inα_{2}with regional climate mirrors the variability in the ratio of mechanical-to-chemical-weathering rates represented byε. The potential importance of these factors has been overlooked by existing studies, and requires further study.

Acknowledgments

[22] Several anonymous reviewers contributed to an improved manuscript. Support for this research was provided by Dartmouth College.