Abstract
 Top of page
 Abstract
 1. Introduction
 2. Data Sets
 3. Review of Generalized AtStation HG
 4. Systematic Downstream Variation in Channel Planform Geometry and CrossSectional Asymmetry: The Effect of ScaleDependent Fluvial Instability
 5. Connection Between Meandering Channel Morphometry and AtStation Hydraulic Geometry
 6. Insights From Contrasting the Statistical and Physical Theories: The Role of Thresholds
 7. Summary and Conclusions
 Appendix A:: Derivation of Meander Bend Properties (Sinuosity, Meander Wavelength, and Radius of Curvature) From HighResolution Hydrography Data
 Appendix B:: Implementation of Linear Theory of River Meandering for Derivation of “Synthetic” Hydraulic Geometry
 Acknowledgments
 References
[1] The power law dependencies between channel morphology and river flows, known as atstation hydraulic geometry (HG), have been recently shown to have exponents that systematically vary with scale (contributing area). To explain these empirical trends, a generalized HG model whose parameters are explicit functions of scale was derived by Dodov and FoufoulaGeorgiou [2004], based on a statistical multiscaling formalism. In this paper we attempt to provide a physical explanation for this scale dependence. The hypothesis we pose is that it arises from the scale dependence of fluvial instability, which induces systematic variation in river planform geometry (e.g., sinuosity, meander wavelength, and radius of curvature) and consequent variations in channel crosssectional shape with scale. In other words, we postulate that the scaledependent HG is a direct consequence of the systematic increase of channel crosssectional asymmetry over reaches of increasing scale. To test this hypothesis, we employ both a direct analysis of observations and also a physical model of meandering rivers, which is based on linearization of the fully coupled equations of mass and momentum balance for water and sediment. We show that the HG emerging from this physical model is scaledependent and agrees with the empirical observations and the statistical multiscaling model.
1. Introduction
 Top of page
 Abstract
 1. Introduction
 2. Data Sets
 3. Review of Generalized AtStation HG
 4. Systematic Downstream Variation in Channel Planform Geometry and CrossSectional Asymmetry: The Effect of ScaleDependent Fluvial Instability
 5. Connection Between Meandering Channel Morphometry and AtStation Hydraulic Geometry
 6. Insights From Contrasting the Statistical and Physical Theories: The Role of Thresholds
 7. Summary and Conclusions
 Appendix A:: Derivation of Meander Bend Properties (Sinuosity, Meander Wavelength, and Radius of Curvature) From HighResolution Hydrography Data
 Appendix B:: Implementation of Linear Theory of River Meandering for Derivation of “Synthetic” Hydraulic Geometry
 Acknowledgments
 References
[2] Hydraulic geometry (HG) refers to the power laws relating stream width W, average depth D, and mean velocity V to discharge Q: W = aQ^{b}, D = cQ^{f}, V = kQ^{m} [Leopold and Maddock, 1953]. These relationships have been observed to hold either for different discharges at an individual cross section (referred to as atstation HG), or for different downstream locations related through some characteristic discharge, e.g., mean annual discharge (referred to as downstream HG). This paper is concerned with the atstation HG.
[3] In a recent paper, Dodov and FoufoulaGeorgiou [2004] presented empirical evidence that the exponents of atstation HG systematically depend on scale, i.e., drainage area upstream (A), and showed that this empirical trend can be captured by a multiscaling formalism of hydraulic geometry factors. Specifically, they postulated and confirmed via analysis of observations that the probability distributions of discharge Q and crosssectional area C_{A} remain statistically invariant under proper rescaling with a random function which depends on scale only (notion of multiscaling). As a result, lognormal multiscaling models were fitted to Q and C_{A} and revised atstation HG relationships (i.e., relationships between C_{A} and Q and V and Q) whose coefficients were explicit functions of scale, were derived. These relationships were called generalized HG and were tested on 85 stations in Oklahoma and Kansas with good agreement to observations.
[4] In this paper, we attempt to provide a physical explanation of the empirically observed and statistically described scale dependence of atstation HG in terms of downstream variations in fluvial instability. First, we briefly review the multiscaling formalism of HG that gives rise to generalized atstation HG relationships. Then, we present an analysis of fluvial instability [Parker, 1976] as a function of contributing area to show that channel planform geometry (e.g., sinuosity, curvature and wavelength) and, particularly, the transition between straight and meandering channels, are scaledependent. To relate channel planform geometry and channel shape, we use the model of Johannesson and Parker [1987, 1989] to calculate the bed topography of representative meander bends of a given Strahler order, and then, the HG of these bends. This model is based on a small perturbation approach which linearizes the governing equations maintaining full coupling between the flow field, bed load transport and bed topography. We show that the atstation HG that emerges from this physical model is scaledependent and agrees with the empirical trends and the proposed statistical model. We also show by direct analysis of observations that the velocity HG exponent depends inversely on channel sinuosity and that sinuosity increases as a function of scale. These results together with findings from the physical model are interpreted as evidence that the physical origin of the scaledependent HG is the systematic increase of channel asymmetry downstream induced by scaledependent fluvial instability.
3. Review of Generalized AtStation HG
 Top of page
 Abstract
 1. Introduction
 2. Data Sets
 3. Review of Generalized AtStation HG
 4. Systematic Downstream Variation in Channel Planform Geometry and CrossSectional Asymmetry: The Effect of ScaleDependent Fluvial Instability
 5. Connection Between Meandering Channel Morphometry and AtStation Hydraulic Geometry
 6. Insights From Contrasting the Statistical and Physical Theories: The Role of Thresholds
 7. Summary and Conclusions
 Appendix A:: Derivation of Meander Bend Properties (Sinuosity, Meander Wavelength, and Radius of Curvature) From HighResolution Hydrography Data
 Appendix B:: Implementation of Linear Theory of River Meandering for Derivation of “Synthetic” Hydraulic Geometry
 Acknowledgments
 References
[10] Evidence was presented by Dodov and FoufoulaGeorgiou [2004], based on 85 stations located in a geologically and climatologically homogeneous region in Oklahoma and Kansas, that the parameters (exponents and preexponents) of atstation HG have a systematic dependence on scale. For example, the atstation HG for velocity V = kQ^{m} was empirically fitted to the 85 stations and estimates of the parameters k and m were plotted as a function of scale A (see points in Figure 3). In the same figure estimates of the parameter m for Sangamon river basin (approximately 500 km eastnortheast from Osage river basin) as given by Stall and Fok [1968] are also shown. A trend of decrease of the exponent m with scale is apparent, indicating an increase in the “linearity” of catchment response as the contributing area increases (i.e., less dependence of velocity on discharge as m tends to zero). At the same time, it was observed that the exponents of downstream HG depend on the frequency of discharge [see Dodov and FoufoulaGeorgiou, 2004, Figure 7].
[11] To explain these empirical findings, a multiscaling framework was proposed within which processes whose spatial variability changes as a function of scale and frequency can be concisely and parsimoniously described. Specifically, it was postulated that the discharge Q and channel crosssectional area C_{A} obey multiscaling lognormal (MSL) models whose pth quantiles are of the form
where α_{(.)}, β_{(.)}, γ_{(.)} and δ_{(.)} are parameters, z_{p} is a standard normal quantile, A is the contributing area, and A_{l}^{0} and A_{l}^{1} represent some limiting contributing areas within which the scaling behavior holds (see Dodov and FoufoulaGeorgiou [2004] and also the original papers of Gupta and Waymire [1990] and Gupta et al. [1994] for more details on the multiscaling theory). The above multiscaling models were fitted to the 85 stations in Oklahoma and Kansas (see Dodov and FoufoulaGeorgiou [2004] for the details of fitting; 10 quantiles of the PDFs of Q and C_{A} of all 85 stations were fitted simultaneously through least squares) and the parameters of the multiscaling models of (1a)–(1b) were estimated and are shown in Table 1.
Table 1. Parameters of the Fitted Lognormal Multiscaling Models for CrossSectional Area and Discharge^{a}Hydraulic Geometry Factors  Estimated Model Parameters 

α  β  γ  δ 


D&FG, 85 Stations 
Crosssectional area, C_{A}  −3.18  0.61  0.84  0.1130 
Discharge, Q  −5.54  0.81  2.61  0.0012 

This Study, 80 Stations 
Crosssectional area, C_{A}  −3.44  0.69  1.11  0.0581 
Discharge, Q  −6.00  0.90  3.00  −0.0721 
[13] In Figure 3 the theoretical curves Φ_{V}(A) and Ψ_{V}(A) computed from equations (3a)–(3d) using the multiscaling parameters of Table 1 are plotted. We see that these theoretical curves are in good agreement with the observations validating thus the proposed multiscaling model for HG.
4. Systematic Downstream Variation in Channel Planform Geometry and CrossSectional Asymmetry: The Effect of ScaleDependent Fluvial Instability
 Top of page
 Abstract
 1. Introduction
 2. Data Sets
 3. Review of Generalized AtStation HG
 4. Systematic Downstream Variation in Channel Planform Geometry and CrossSectional Asymmetry: The Effect of ScaleDependent Fluvial Instability
 5. Connection Between Meandering Channel Morphometry and AtStation Hydraulic Geometry
 6. Insights From Contrasting the Statistical and Physical Theories: The Role of Thresholds
 7. Summary and Conclusions
 Appendix A:: Derivation of Meander Bend Properties (Sinuosity, Meander Wavelength, and Radius of Curvature) From HighResolution Hydrography Data
 Appendix B:: Implementation of Linear Theory of River Meandering for Derivation of “Synthetic” Hydraulic Geometry
 Acknowledgments
 References
[14] Channel crosssectional shape can be easily shown to have a significant effect on the HG of an idealized channel. For example, consider two channels: one with wide rectangular and the other with triangular (asymmetric or symmetric) cross section. Using Manning's relation for velocity V ∼ C_{A}^{2/3}P^{−2/3}, (for the same Manning's n and channel slope, where C_{A} is the crosssectional area and P is the wetted perimeter) it can be shown that the HG exponent for velocity is 0.4 for the wide rectangular channel and 0.25 for the triangular channel. Considering that the transition from a straight to a meandering channel consists of a combination of almost rectangular to trapezoidal to asymmetric triangular cross sections (e.g., see Johannesson and Parker [1987, 1989] for details), and the observation that meandering induces asymmetry in the cross sections of natural rivers (as evidenced by Leopold and Wolman [1960], who site that approximately 90% of meandering channels have asymmetric cross sections), a possible connection seems to emerge as to whether the scale dependence of HG is due to the systematic increase of channel asymmetry downstream induced by scale dependence in river planform geometry. In particular, we pose and answer the following two questions: (1) how and why does the sinuosity of meandering rivers change with scale, and (2) how does the degree of sinuosity affect the channel cross section asymmetry and in general the HG? The first question is addressed in this section and the second question in section 5.
[15] The transition between the three characteristic fluvial morphologies of channels (meandering, braiding, and straight) has been considered by Parker [1976] based on stability analysis of a twodimensional alluvial river model previously proposed by Hansen [1967] and Callander [1969]. The approach is based on a perturbation technique, involving a small parameter representing the ratio of sediment transport to water transport. Parker [1976] derived an instability criterion ɛ*
where S is the alongchannel slope and W, Fr and D are respectively the width, the Froude number and the mean depth at formative discharges (usually assumed bankfull ones), which can be used to differentiate between the three regimes. The theory indicates that if the depth to width ratio D/W ≪ 1 and sediment transport is present (conditions which are usually satisfied in natural rivers) a tendency toward either meandering or braiding exists. Meandering occurs for S/Fr ≪ D/W, braiding occurs for S/Fr ≫ D/W and transition between the two regimes occurs if S/Fr ≈ D/W. Thus if ɛ* ≪ 1 meandering occurs, if ɛ* ≫ 1 braiding occurs (with a number of braids n_{br} ∼ ɛ* ) and if ɛ* = O(1) transition between meandering and braiding is present. The condition for the maintenance of straight channels is assumed to be D/W > 0.1.
[16] In the work of Parker [1976] no consideration of scale was made and inferences, about flow instabilities and transition from one regime to another, were made irrespectively of whether channels drained a small or a large upstream area. In our work, we use the available data (data sets A and B) to perform a stability analysis as a function of scale.
[17] In order to apply stability analysis, we first need to determine the bankfull discharge Q_{bf}, width W_{bf}, average depth D_{bf} and mean velocity V_{bf} for all available stations based on the empirical stagedischarge and widthdischarge relationships. Bankfull conditions were considered to occur when a break in slope was observed in both stagedischarge and widthdischarge relationships (a transition to a very slow increase in stage and very rapid increase in width after bankfull discharge is reached). An example of determination of bankfull discharge is given in Figure 4. The width, mean velocity, mean depth and crosssectional area at bankfull were obtained from the nearest measurement below the break. In some cases, where Q_{bf} was determined using only stage and discharge data (i.e., no data for W, V, etc. were available at around bankfull discharges), W_{bf}, V_{bf} and D_{bf} were obtained by eye from the corresponding loglog plots versus discharge Q. The bankfull channel geometries and discharge for the 92 stations of data set A (for which these properties were determined) are presented in Figure 5 as a function of contributing area (scale). In the same figure, the atstation local channel slopes S and the ratio D_{bf}/W_{bf} needed for stability analysis are also given as a function of scale.
[18] The observations (points) in Figure 5 suggest some trends in the bankfull properties with scale. Specifically, it is noted that W_{bf}, D_{bf} and Q_{bf} versus scale can be well described by double loglog linear relationships which exhibit a break at a scale of approximately 700 km^{2}. The solid lines shown for these variables in Figure 5 were fitted by eye keeping the scale of transition from one loglog linear relationship to the other equal to 700 km^{2} as clearly dictated by the W_{bf}, D_{bf} and Q_{bf} relationships. The dashed lines in the V_{bf} and D_{bf}/W_{bf} versus area plots were derived from the fitted lines of W_{bf}, D_{bf} and Q_{bf}. The solid line in the relationship of alongchannel slope S versus contributing area was fitted by eye with no assumed break using the available atstation local slopes and the information extracted from the Digital Elevation Models (DEM) of Neosho and Osage river basins.
[19] Once we have obtained the channel morphometry parameters at bankfull as a function of scale we can proceed with the stability analysis described above. In Figure 6a we plot the ɛ* criterion at bankfull computed from (4) versus scale for the 64 stations of data set A for which the atstation alongchannel slopes were available. The fact that ɛ* is almost constant for scales up to ∼700 km^{2} suggests that sinuosity should be almost constant for streams that drain areas up to that scale. However, the plot of S/Fr_{bf} versus D_{bf}/W_{bf} in Figure 6b shows that for medium to small scales (∼700 km^{2} to 10 km^{2}) the ratio D_{bf}/W_{bf} approaches values greater than 0.1 (from below) implying transition from moderate meandering to straight channels as the contributing area decreases. In contrast, for scales larger than ∼700 km^{2}, ɛ* decreases with contributing area which suggests increase in fluvial instability with scale. At the same time, the plots of D_{bf}/W_{bf} versus contributing area and S/Fr_{bf} versus D_{bf}/W_{bf} show that the ratio D_{bf}/W_{bf} is almost constant with scale implying suppressed instability for scales of order larger than O(1000 km^{2}). It is noted that the broken line in Figure 6a and the arrows in Figure 6b are computed from the loglinear approximations in Figure 5.
[20] Considering the fact that the sinuosity can be used as a descriptor of fluvial instability for ɛ* ≪ 1, the overall inference from the above analysis is that moderate sinuosity should be observed at scales of order O(1000 km^{2}), a trend for slight decrease of sinuosity with increase in contributing area should be present at larger scales, and almost straight channels (sinuosity converging to 1) should dominate at scales less than approximately 100 km^{2}.
[21] To independently check the above conclusion, we performed an analysis of the highresolution hydrography data set for the Neosho and Osage river basins (data set B). The data set was first carefully examined to clean all loops and artificial channels (where possible) in order to assign a particular Strahler stream order to every channel in the river network. Then the sinuosity, meander wavelength and channel slope were computed separately for every bend using simultaneous analysis of vector and elevation raster data. (Technical details about the definition of the meander bend and the estimation of sinuosity, meander wavelength, minimum radius of curvature for a given channel bend are given in Appendix A).
[22] The analysis of planform geometry showed that the distributions of the above parameters for a given stream order are nonsymmetric (e.g., approximately lognormal for meander wavelength and power law distribution for sinuosity, e.g., see an example in Figure 7). This is why, in our analysis it was considered more appropriate to use medians rather than means as representative estimates of these parameters:Median meander wavelength
Median sinuosity
where λ_{ω}^{i} and s_{ω}^{i} are respectively the wavelength and sinuosity of an individual meander bend of order ω.
[23] If a meander bend is assumed to follow a sinegenerated curve (a common assumption in fluvial geomorphology), approximate estimates of other descriptors of meander planform geometry can be derived. These parameters (used in our analysis later on) are the minimum radius of curvature of the bend R_{min} ≈ λ [(s − 1)/s]^{−1/2}/4.4π and its angle amplitude Θ_{O} ≈ 2.2[(s − 1)/s]^{1/2} (see Langbein and Leopold [1966] and Johannesson and Parker [1989] for details). As could be expected, for bends of a given order the minimum radius of curvature and the angle amplitude also followed nonsymmetric distributions and, respectively, their median values were considered representative values of these parameters:Median angle amplitude
Median of the minimum radius of curvature
[24] The estimated values of the parameters in (5a)–(5b) and (6a)–(6b) are given in Table 2 grouped according to channel's stream order and are also plotted in Figure 8. Note that channel stream order relates monotonically to contributing area as shown in Figure 8d and therefore, area and stream order are used interchangeably in the rest of the paper.
Table 2. Median Parameters of Channel CrossSectional and Planform Geometry for Meander Bends of Order ω Used as an Input to the Model of Johannesson and Parker [1987, 1989]^{a}Order ω  Area A, km^{2}  Mean Velocity V_{bf}, m s^{−1}  Mean Depth D_{bf}, m  Channel Slope S  Halfwidth b_{bf}, m  Wavelength λ, m  Perturbation Parameter ψ_{0}  Sinuosity s  Angle Amplitude Θ_{0}, rad  Minimum Radius of Curvature R_{min}, m  Number of Bends 


1  0.4  0.95  0.61  0.01286  1.00  112.5  0.034  1.09  0.62  24.7  161,219 
2  2.8  0.93  0.95  0.00690  2.41  129.7  0.090  1.15  0.80  24.0  44,618 
3  11.6  0.92  1.31  0.00436  4.58  175.0  0.136  1.19  0.89  31.2  16,769 
4  52.9  0.90  1.84  0.00266  9.15  258.9  0.193  1.22  0.94  45.3  5562 
5  220.9  0.89  2.55  0.00167  17.54  442.0  0.226  1.26  1.00  74.2  1007 
6  989.4  0.90  3.44  0.00103  30.69  604.7  0.306  1.30  1.05  97.4  983 
7  7087.2  1.06  4.27  0.00054  37.30  1018.5  0.208  1.25  0.98  168.7  692 
8  14,581.2  1.13  4.62  0.00043  40.06  1848.0  0.116  1.26  1.00  305.2  31 
[25] The results clearly support the prediction from stability analysis, i.e., maximum (but moderate, 1.30) median sinuosity is observed at streams of order 6 (∼1000 km^{2}), decreasing median sinuosity with an increase of scale to streams of order 8 (≈10,000 km^{2}) and approach to almost straight channels (median sinuosity 1.09) at streams of order 1 (≈1 km^{2}). This agreement between empirical observations and physical theory is satisfying and provides insight into the physical origin of the empirically observed scaledependent sinuosity in natural rivers (Figure 8c) in terms of fluvial instability (Figure 6).
[26] Although the analysis of fluvial instability (based on the variation of channel properties with scale) supports the increase of channel sinuosity in the downstream direction, there is another factor that strongly affects channel crosssectional shape and planform geometry, namely, the sediment supply regime (see Leopold and Wolman [1957], Schumm and Khan [1972], and Ikeda [1989], among others). Particularly, in their paper Schumm and Khan [1972] reported results from flume experiments in which they observed that a change in the suspended sediment load rapidly increased sinuosity and transverse slope of the experimental channel. This observation suggests that if there is a trend of increase in channel sinuosity with scale, such a trend might be related to a change in sediment supply regime in the downstream direction, the existence of which would be an additional support to our conclusions. To check if the sediment supply regime does change with contributing area we analyzed the suspended sediment concentration at different flow conditions for the 92 stations of data set C. In Figure 9 we plot the suspended sediment concentration corresponding to frequency of exceedance of daily discharges 50% and 5% in order to consider two quantiles below bankfull conditions. It is seen that both quantiles follow approximately straight lines with slopes respectively 0.29 and 0.36 in loglog space. This overall trend of an increase in suspended sediment concentration with scale for below bankfull conditions is, according to Schumm and Khan [1972], another physically based motivation to assume that the sinuosity increases in the downstream direction.
[27] Having established the dependence of channel planform geometry on scale, in the next section we continue with the next important step in our analysis, namely, to provide a connection between the increasing sinuosity and the parameters of atstation hydraulic geometry. It is noted that establishing a direct (observationbased) connection between channel sinuosity and channel cross section asymmetry is practically infeasible as channel cross section data over many reaches with different contributing areas do not exist. Such a connection can only be inferred using a physical model of water and sediment transport, and this is the approach we followed as described in the next section.
5. Connection Between Meandering Channel Morphometry and AtStation Hydraulic Geometry
 Top of page
 Abstract
 1. Introduction
 2. Data Sets
 3. Review of Generalized AtStation HG
 4. Systematic Downstream Variation in Channel Planform Geometry and CrossSectional Asymmetry: The Effect of ScaleDependent Fluvial Instability
 5. Connection Between Meandering Channel Morphometry and AtStation Hydraulic Geometry
 6. Insights From Contrasting the Statistical and Physical Theories: The Role of Thresholds
 7. Summary and Conclusions
 Appendix A:: Derivation of Meander Bend Properties (Sinuosity, Meander Wavelength, and Radius of Curvature) From HighResolution Hydrography Data
 Appendix B:: Implementation of Linear Theory of River Meandering for Derivation of “Synthetic” Hydraulic Geometry
 Acknowledgments
 References
[28] To study the possible connection between channel planform geometry and atsite HG exponents, we first need to define “representative reaches” upstream of each station. It is noted that the length of the representative reach assigned to each station will depend on the Strahler order of the stream (or the area draining to each station) with stations of, say, order 3 being assigned a smaller representative reach than stations of order 5, and so forth. It is desirable that, on the average, the number of meanders encountered in each representative reach is the same. To accomplish this we used the median planform properties of streams of different orders as presented in Table 2 (e.g., median sinuosity s_{ω} and median wavelength λ_{ω} of stream of order ω) and computed for each order ω the alongvalley length of a “median meander bend” as λ_{ω}/s_{ω}. The alongvalley length of the representative reaches was chosen to be approximately 3 times the alongvalley length of the median meander bends, i.e., a length of approximately 3λ_{ω}/s_{ω} for a stream of order ω, measured upstream from each station along the valley slope. Since the spatial extent of data set A is much larger than the highresolution hydrography data of data set B (which was used for the extraction of planform properties of Table 2) additional work was performed to extract planform properties in the regions lacking highresolution hydrography data. Namely, the channel center line in the vicinity of every station (except the ones in Neosho and Osage river basins – data set B) were digitized from 1:24,000 topographic maps over lengths of approximately 3λ_{ω}/s_{ω} according to the order of the channel. Then, the atstation channel sinuosity and average curvature were computed from the digitized lines.
[29] In Figure 10, we plot the atstation HG exponents for velocity versus the atstation sinuosity and normalized channel curvature (W_{bf}^{i}^{i}, where W_{bf}^{i} is the bankfull width and ^{i} the average curvature of the representative reach of station i) for the 96 stations of data set A. Clearly, both plots show a dependence of HG on channel planform geometry expressed in terms of a decrease of the exponent m with an increase in sinuosity and normalized curvature. As we have already shown that the sinuosity increases in the downstream direction, the immediate conclusion is that the overall trend in the region of interest is toward a decrease of velocity exponents with an increase in contributing area.
[30] To show that the dependence between HG exponents and sinuosity is due to the increasing channel asymmetry downstream we employed the linear model of meandering rivers of Johannesson and Parker [1987, 1989] (see Appendix B for a brief description of the model). Johannesson and Parker [1987, 1989] proposed a twodimensional (longitudinal and transverse coordinates) model for river meandering based on momentum balance, continuity and sediment conservation equations under the assumption of steady slender flow. This model calculates the flow field and bed topography in curved channels with an erodible bed. The coupling between flow field, bed load transport and bed topography was shown in that work to cause significant increase of the lateral bed slope which was not reproduced by previous studies neglecting the convective transport of primary flow momentum by secondary flows in the transverse direction. The governing equations of the model are summarized in Appendix B.
[31] The model of Johannesson and Parker [1987] was adopted in our analysis by using it on representative meander bends of various orders ω. By supplying channel bankfull properties for a channel of order ω, i.e., width W_{bf}^{ω}, average depth D_{bf}^{ω}, average velocity V_{bf}^{ω}, channel slope S^{ω}, meander wavelength λ_{ω}, angle amplitude Θ_{O}^{ω} and a perturbation parameter Ψ_{O} defined as Ψ_{O} = πΘ_{O}^{ω}W_{bf}^{ω}/λ_{ω}, we can compute, using the model, the bed topography at several cross sections of that meander bend. By using properties of meandering channels of different orders ω (extracted from the region of NeoshoOsage river basins in Oklahoma and Kansas), regional HG relationships can be derived as a function of scale and compared with the empirical ones.
[32] The representative meander bends were chosen to have crosssectional geometry represented by the power law approximations in Figure 5 (data set A) and median planform geometry extracted from the highresolution hydrography of Neosho and Osage river basins (data set B). In analogy to (6a)–(6b), the perturbation parameter was computed asMedian perturbation parameter
[33] A summary of all the needed parameters for constructing representative meander bends of different orders is given in Table 2 (except the median size of bed material D_{50}, which was assumed to be 1.5 mm, the average for the region of interest, since no trend of downstream fining was observed). Using these parameters in the governing equations of the physical model, the steady state bed topography of representative meander bends was computed for different order streams.
[34] In Figure 11, example cross sections of synthetic bends of streams of order 2 and 7 are shown. Clearly, the trend is toward an increase in transversal slope, and respectively increase in cross section asymmetry with an increase in stream order. Furthermore, the percentage of the alongchannel length (or the number of cross sections) significantly affected by asymmetry increases considerably with the stream order. The cross sections with a break in transversal slope (e.g., see Figure 11 for bends of order 7) are those at which the perturbation in channel depth exceeds the mean depth of the channel. In such cases we assume that the secondary currents scour additionally the outer bank, depositing sediment at the inner 2/3 of the channel width, while at the same time preserving the width, mean depth and mean transversal slope of the channel as predicted by the theory (see also Appendix B).
[35] Having the channel cross sections of each synthetic meander bend, a conditional HG was extracted based on the assumption of normal flow at any cross section in the bend and using the ManningStrickler's power relation for resistance U/u_{*} = 8.1(H/k_{S})^{1/6}, where u_{*} = is the shear velocity and k_{S} the effective roughness (estimates of effective roughness were computed for every channel order using this relation and bankfull channel properties given in Table 2, i.e., substituting H = D_{bf}^{ω}, and U = W_{bf}^{ω}, and solving for k_{S}^{ω}). The HG is called conditional since it represents the expected velocity through the bend, conditional on a given discharge at every cross section as the discharge increases.
[36] Examples of conditional hydraulic geometry for velocity are given in Figure 12 for synthetic meander bends of order 1, 3, 5 and 7. Note that the theoretical HG relationships are slightly nonlinear in the loglog domain as was pointed out by Ferguson [1986]. The purpose of their approximation by power laws (the straight lines in Figure 12) in this study is to facilitate the comparison with empirical data in terms of a few parameters, i.e., the preexponents k and the exponents m in these relationships.
[37] In Figure 13 we plot the coefficients k and the exponents m of the predicted (from linear theory) and the empirical atstation HG for velocity as a function of scale (for data set A). For comparison, the predictions of HG based on the multiscaling models of discharge and crosssectional area proposed by Dodov and FoufoulaGeorgiou [2004] and reviewed in section 2 are also given (i.e., as computed from relationships (3c) and (3d) using the parameters of Table 1 fitted to data set A). What is evident from these figures is that independently of the functional type of the decay of HG exponents with scale, the trend of decreasing magnitude of these parameters with scale is well reproduced by both the multiscaling model and the linear theory.
[38] To test the statistical significance of the trend we apply a hypothesis testing with the null hypothesis that the slope of the least squares fit to the relationship m versus ln(A) is equal to zero. The statistic for this test (under the premise of Gaussian error model [e.g., see Devore and Peck, 1996]) is given by the expression
where b is the slope of the linear model fit, _{i} is the estimate of the linear model fit for a given ln A_{i} and n is the number of samples. The computed test statistics is t = −5.06 and for 94 degrees of freedom (96 stations – 2), the probability P(t < −5.06) and P(t > 5.06) is 2.04 × 10^{−4}% (double sided test to check if b ≠ 0). Since this percent is much less than any reasonable confidence threshold the hypothesis is rejected, i.e., the trend of decrease of exponent m with scale is statistically significant. Although visually minor, this trend has been shown to play an important role when considered in the context of hydrologic response [see Dodov and FoufoulaGeorgiou, 2004] since it significantly affects the shape of the hydrograph and implies different degrees of nonlinearity in catchment response at different scales.
6. Insights From Contrasting the Statistical and Physical Theories: The Role of Thresholds
 Top of page
 Abstract
 1. Introduction
 2. Data Sets
 3. Review of Generalized AtStation HG
 4. Systematic Downstream Variation in Channel Planform Geometry and CrossSectional Asymmetry: The Effect of ScaleDependent Fluvial Instability
 5. Connection Between Meandering Channel Morphometry and AtStation Hydraulic Geometry
 6. Insights From Contrasting the Statistical and Physical Theories: The Role of Thresholds
 7. Summary and Conclusions
 Appendix A:: Derivation of Meander Bend Properties (Sinuosity, Meander Wavelength, and Radius of Curvature) From HighResolution Hydrography Data
 Appendix B:: Implementation of Linear Theory of River Meandering for Derivation of “Synthetic” Hydraulic Geometry
 Acknowledgments
 References
[39] A closer examination of Figure 13 suggests some differences in the trends of the exponent m with contributing area predicted from the statistical and physical theories, i.e., for the linear theory a steep decay up to approximately 700 km^{2} and almost constant exponents after that, and for the multiscaling theory exponents following almost a straight line with logarea. To explain this difference, let us consider the plots in Figure 14. In these plots we compare the scaling of bankfull discharges Q_{bf} and crosssectional areas = W_{bf}D_{bf} of data set A with the scaling of the empirical quantiles of Q and C_{A} corresponding to frequencies of exceedance 50% and 5%. The quantiles of Q and C_{A} were computed directly from observations and nonparametrically, i.e., without assuming any particular distribution for daily discharges and any particular relationship (e.g., power laws) between discharge and crosssectional area.
[40] It is evident from Figure 14 that the below bankfull empirical logquantiles of Q and C_{A} (points) follow almost straight lines with logarea, and are in good agreement with the quantiles predicted by the multiscaling theory (solid lines). This observation supports once again the multiscaling formalism for below bankfull discharges introduced by Dodov and FoufoulaGeorgiou [2004]. In that study only quantiles corresponding to frequencies of nonexceedance of Q and C_{A} of up to 99.4% were explicitly considered in the multiscaling model fitting (specifically, ten frequencies of nonexceedance p = {0.006, 0.026, 0.082, 0.202, 0.391, 0.609, 0.798, 0.918, 0.974, 0.994}), thus neglecting the effect of nearbank and overbank flows whose frequency of nonexceedance is typically larger than 99%. In contrast, hydraulic geometry derived from the linear theory is based on the double power law approximations of channel bankfull morphometry shown in Figure 5 and thus explicitly incorporates a physical threshold of the system and its dependence on scale (i.e., the transition from below to above bankfull conditions and how this transition changes with scale). This physical threshold is not accounted for in the multiscaling theory of HG. Despite the above explained disagreement between the multiscaling theory and physical theory predicted HG, one could argue that both predictions are acceptable within the scatter of empirical observations, which is considerable especially for large contributing areas (see Figure 13).
[41] It is worth commenting here that the scaling and scaling break of above bankfull flows seen in Figure 14 (left) is expected to be reflected in the scaling of (maximum annual) floods. For river reaches with contributing areas less then 700 km^{2}, the frequency of occurrence of overbank flows is almost constant (i.e., parallel lines of Q_{bf}, Q_{50%} and Q_{95%}, and and in Figure 14), while for larger rivers this frequency seems to depend on contributing area (i.e., different slopes of Q_{bf} compared to Q_{50%} and Q_{95%}, and compared to and In other words, river reaches that drain large areas will be more frequently flooded than smaller ones. Considering the retardation effects of overbank storages on peak flows, one could easily guess that the peak flow statistics will be affected in a different way at different scales with a trend of decreasing variability downstream. This observation is revealing and is analyzed in detail in a subsequent publication (B. Dodov and E. FoufoulaGeorgiou, Fluvial processes and streamflow variability: Interplay in the scalefrequency continuum and implications for scaling, submitted to Water Resources Research, 2004, hereinafter referred to as Dodov and FoufoulaGeorgiou, submitted manuscript, 2004) in the context of the socalled multiscaling theory of flood peaks [see Gupta and Waymire, 1990; Gupta et al., 1994; Gupta and Dawdy, 1995].
7. Summary and Conclusions
 Top of page
 Abstract
 1. Introduction
 2. Data Sets
 3. Review of Generalized AtStation HG
 4. Systematic Downstream Variation in Channel Planform Geometry and CrossSectional Asymmetry: The Effect of ScaleDependent Fluvial Instability
 5. Connection Between Meandering Channel Morphometry and AtStation Hydraulic Geometry
 6. Insights From Contrasting the Statistical and Physical Theories: The Role of Thresholds
 7. Summary and Conclusions
 Appendix A:: Derivation of Meander Bend Properties (Sinuosity, Meander Wavelength, and Radius of Curvature) From HighResolution Hydrography Data
 Appendix B:: Implementation of Linear Theory of River Meandering for Derivation of “Synthetic” Hydraulic Geometry
 Acknowledgments
 References
[42] The relations between channel morphology and discharge (known as hydraulic geometry: HG) are widely used by geomorphologists and hydrologists since their introduction by Leopold and Maddock [1953]. Recently, Dodov and FoufoulaGeorgiou [2004] showed that these relationships might deviate from simple power laws with constant exponents as initially proposed. More specifically, they showed that the exponents of the atsite HG systematically depend on upstream contributing area. The authors provided empirical evidence for this scale dependence and proposed a statistical framework (based on a multiscaling formalism) within which these empirical trends could be interpreted and quantified.
[43] In the present paper, we offer a physical explanation of the scale dependence of HG. Using a physically based theory which connects channel planform geometry (e.g., sinuosity, curvature and meander wavelength) and channel crosssectional shape under preservation of the momentum, water and sediment [Johannesson and Parker, 1987, 1989], we predict HG and show that the physically derived HG agrees with the empirical trends and to a degree with the statistically predicted HG based on the multiscaling theory. This agreement, together with a parallel analysis of the dependence of channel planform geometry on scale and dependence of atstation HG exponents on sinuosity, is interpreted as evidence that the physical origin of the scaledependent HG is the systematic increase of channel asymmetry downstream induced by scaledependent fluvial instability.
[44] It is noted that the systematic change of HG exponents with scale reported herein has been extracted from thousands of streams in a region with overall homogeneous geologic, topographic and hydroclimatic conditions, and thus, for every scale of interest, it reflects the spatially variable planform and channel geometry characteristics present in that region (e.g., see PDFs of sinuosity for a particular scale in Figure 7). In this sense, it is noted that the notion of scaling of atsite HG is to be interpreted in statistical terms (i.e., akin to the scaling and regionalization of floods) and not as the change of HG as one goes downstream along individual reaches of the river.
[45] An interesting observation, namely the dependence of the frequency of occurrence of overbank flow on contributing area, suggests that river reaches that drain large areas are more frequently flooded than smaller ones. This observation also sheds light into the decreasing variability of flood peaks for large areas (considering the retardation effect of overbank storages on peak flows) and is explored in its own right in a subsequent paper (Dodov and FoufoulaGeorgiou, submitted manuscript, 2004) in the context of the multiscaling theory of flood peaks [see Gupta and Waymire, 1990; Gupta et al., 1994; Gupta and Dawdy, 1995].
[46] The development of synthetic HG (used in this paper to physically explain the systematic variations of atstation HG with scale) can be itself a valuable practical tool for runoffrouting applications in ungauged basins. A methodology based on the linear theory of river meandering can be used for computing synthetic HG of individual channels (based on supplied representative values or PDFs of the parameters describing channel planform geometry and crosssectional morphometry), which can then be applied for distributed probabilistic routing in ungauged catchments. Such an approach would provide realistic bankfull discharges and timely transition to overbank regime, which is very important from both hydrologic and geomorphologic point of view.