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Evaluating computational models of braided rivers
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(final submission version, Aug 2003)
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Andrea B. Doeschl1,*, Peter E. Ashmore2 and Matt Davison1
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The University of Western Ontario
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London, Ontario, Canada
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N6A-5B7
Department of Applied Mathematics
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The University of Western Ontario
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London, Ontario, Canada
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N6A-5C2
Department of Geography
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*contact author – current address:
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School of GeoSciences
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The King’s Buildings, University of Edinburgh
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Edinburgh, UK, EH9 3JG
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(andrea.doeschl@ed.ac.uk)
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Phone: +44 – 0131 – 535 4110
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Doeschl et al.
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Abstract
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In recent years various computational models of braided rivers have been proposed.
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These range from detailed computational fluid dynamics studies designed to predict
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the evolution of a particular braided river to the qualitative cellular automata-based
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model of Murray and Paola, designed to identify in a broad-brush way the most
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important conditions responsible for river braiding. Evaluating models such as
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Murray and Paola’s is challenging as the evaluation cannot be reduced to a direct
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comparison between model output and a particular natural or laboratory river, but
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must involve the identification of characteristic properties common to all braided
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rivers. The Murray-Paola model has provided a valuable service to the braided river
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community by stimulating research aimed at identifying such properties. Existing
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research suggests that the planform characteristics of the Murray-Paola model are
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similar to those of real braided rivers. We propose some new model evaluation
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techniques for additional aspects of braided river morphology and apply them to
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comparisons of a physical model and the Murray-Paola numerical model with a real
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braided river, the Sunwapta River. Our techniques show that the flume river shares
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the statistical characteristics of the braided river. However, even certain planform
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characteristics of the Murray-Paola model seem different from that of a real river.
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The topographic characteristics of the Murray-Paola river are completely different
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from that of the real river. We present this research not to attack the work of Murray
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and Paola, which was not, after all, designed to bear this kind of analysis, but to make
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the point that to evaluate generic braided river models, a variety of statistical metrics
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including topographic ones must be utilized and to provide possible metrics for future
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evaluation.
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Doeschl et al.
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Keywords: Model evaluation, Braided rivers, Scale invariant properties, cellular
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models
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1. Introduction
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During the last few decades, the demand for reliable methods to analyse and predict
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braided river behaviour has grown considerably. Increased flood frequencies at
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braided rivers in China and Bangladesh have caused substantial changes in the river
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planform, which imposed dramatic damages to the inhabitants and to the
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infrastructure of the corresponding regions. These catastrophes demonstrate the urgent
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need for a thorough understanding of braided river dynamics and for the ability to
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forecast the evolution of the planform changes in the short and medium-term future.
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Small-scale laboratory braided river models and computational models play a key role
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in the research of braided river dynamics as they offer the possibility to generate and
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observe the evolution of a braided river model in great detail over a sufficiently long
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time period for known initial and boundary conditions. These models thus reveal
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various aspects of braided rivers that cannot easily be assessed in natural braided
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rivers, such as the constant interaction between planform and topographic
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characteristics and their collective response to variations in hydraulic and geometric
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conditions. As a consequence, an increased awareness that braided river dynamics
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cannot be described by the development of the channel structure independently from
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the development of the topography has emerged (Ashmore, 2000; Murray and Paola,
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1997; Furbish, 2003).
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Doeschl et al.
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However, progress in developing physical and computational models as a base to
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study the dynamic behaviour of braided rivers, as well as the coupled behaviour
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between flow and topographic characteristics, also raises the demand for appropriate
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model evaluation tools that capture these multiple aspects of braided river morpho-
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dynamics. Present quantitative methods for model evaluation concentrate mainly on
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static flow or planform properties, but the evaluation of topography and dynamics is,
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at present, primarily restricted to qualitative assessment (Sapozhnikov et al., 1998;
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Murray and Paola, 1997, 1998; Paola, 2000; Thomas and Nicholas, 2002; Jagers,
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2003). Adequate model evaluation requires quantitative methods that cover multiple
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aspects of the modelled system and, as a prerequisite, a quantitative description of
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these characteristics of natural braided rivers. In this sense, model evaluation has
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much in common with a robust statistical data analysis. The latter does not stop with a
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qualitative visual inspection of the data describing the system. Instead, it uses a
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variety of quantitative techniques to confirm or disregard the hypotheses inferred from
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prior visualization. Essentially the problem is to find quantitative criteria that
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characterize braided rivers and allow modellers to assess the extent to which model
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output reproduces the morphology, dynamics and response to external forcing, of
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‘real’ braiding.
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The first part of this paper focuses on quantitative methods to describe braided river
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characteristics, discussing both existing approaches that have focussed primarily on
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planform and proposing additional methods that would incorporate topographic
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characteristics and morpho-dynamics. In the second part of this paper, some of these
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methods are applied to evaluate the cellular braided river model of Murray and Paola
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(hereafter MP model) (1994). The MP model was intended as a simple tool to show
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that braided channel patterns arise spontaneously from flow over non-cohesive
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sediment and to find the minimal set of ingredients needed to generate a realistic
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braided behaviour but the model has been shown to reproduce many of the planform
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characteristics of natural braided rivers. However, little attention has been paid to
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whether topographic characteristics of natural braided rivers are also reproduced by
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the model and whether the modelled rivers respond correctly to given external
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forcing. The present study extends the model evaluation by simultaneously
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considering planform and topographic patterns and by using data from a laboratory
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flume for comparing the intrinsic spatial and temporal scales of both systems as a
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means to assess their response to external forcing.
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The point of this analysis is not to criticize the MP model, but rather to use it as an
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example with which to demonstrate the need for a larger range of assessment tools
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and the necessity to consider multiple aspects of braided river behaviour
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simultaneously in the model assessment. The latter imposes stricter criteria on a
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model’s validity, but is essential for fully identifying a model’s capacity and, as a
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consequence, for further progress on understanding fundamentals of braiding in the
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exploratory mode, as described by Murray (2003). This can only be achieved by
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critical evaluation of progress at each stage and finding what is being reproduced and
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what is not.
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The suggestions and methods introduced in this paper offer by no means a complete
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solution for robust model evaluation, but should be rather considered as a step
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forward in the quest of adequate approaches in modelling braided river morpho-
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dynamics.
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2. Existing methods to analyse braided river characteristics
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Two main sets of characteristics identify and describe a braided river: the spatial and
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temporal scales that distinguish a particular braided river from another and the scale
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invariant properties common to all braided rivers.
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The characteristic spatial and temporal scales of planform and topographic features
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are the river’s response to the driving hydraulic and geometric conditions. A braided
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river model that aims to improve the understanding of braided river mechanics must
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therefore represent the true relationship between the hydraulic and geometric controls
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and the resulting planform, topography and morpho-dynamics. Although braided
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rivers are characterized by continually unstable and evolving morphology, the overall
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average morphology is statistically stable (Paola, 2000) and this provides a general
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basis for assessing the river morphology and response.
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Whereas the definition of characteristic scales and their functional controls helps to
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distinguish quantitatively one braided river from another, scale invariant properties
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distinguish braided rivers from other systems. These properties express statistical
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similarities between braided rivers of different scales, which result from the same
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underlying mechanical processes inherent in all braided rivers. Examples of scale
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invariant properties in braided rivers may range from a characteristic organisation of
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pool-bar units to the sorting of sediment along a braid bar. A model that claims to
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represent the main mechanical processes of braided rivers must therefore reproduce
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these scale invariant properties.
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2.1. Characteristic spatial and temporal scales
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Characteristics that describe the style or degree of braiding include the braiding index,
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the total or average width, the average confluence-confluence spacing and the total
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sinuosity (for a review of some of these characteristics see Bridge, 1993, Mosley
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1983). Examples of characteristic scales of the topography are scour depth or bar
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height, as well as measures that describe the structural variability in the topography
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(Zarn, 1997). Empirical studies that relate the scales of these features to flow and
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material properties via multiple regression analyses, are invaluable for model testing.
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As Ashmore (2000) pointed out, few empirical formulas are available for braided
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rivers, although Mosley (1983) provides a comprehensive data set on channel
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dimensions and the distribution of flow at different stages that is equivalent to the ‘at-
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a-section’ hydraulic geometry for single thread channels. The large spatial variability
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in characteristics, along with ambiguity or a lack of consensus on measurements and
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definitions, contributes to current uncertainty in defining braiding characteristics. For
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example, does braiding intensity refer to all channels or only those transporting bed
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sediment at high stage and are all channel segments included regardless of length or
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width? Seldom are these characteristics defined over a range of flow conditions or
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even at a well-defined single flow state. In some cases the complete frequency
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distributions rather than mean values may be more diagnostic (Ashmore, 2000) but
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these are difficult to obtain and therefore seldom known.
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An even greater challenge than finding appropriate definitions of spatial scales is the
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definition of characteristic time scales. Identification of characteristic temporal scales
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is important in assessing the rate at which models function and the relationship to
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static spatial characteristics. Without the definition of characteristic temporal scales, it
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is impossible to quantify the dynamics of braided rivers.
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Morphology changes abruptly and locally and adequate characterization of process
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rates in the field may require several years of observation. While rates of
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morphological evolution are tied to rates of bed material transport, the relationship
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between them remains uncertain (Ashmore 2000; Hoey et al., 2000) so that
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measurement of morphological change is required in addition to direct measurement
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of bed load transport rates. At present the rate of morphological development and bed
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sediment transport are known to fluctuate widely at constant total discharge (Ashmore
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and Church, 1995), but average rates also increase with discharge in a given river or,
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experimentally, at higher total stream power for a given grain size (Ashmore, 2000).
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Large spatial features take longer to develop than small features and the life-time of
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larger channels must depend on the life-time of their smaller tributaries. However,
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temporal characteristics of the morpho-dynamics have not been quantified and related
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to spatial characteristics or hydraulic conditions. Characteristic time scales of braided
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rivers may be defined in various ways. Possibilities include the time that the system
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takes to establish its statistical equilibrium or the average time of existence of channel
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confluences and diffluences (Paola, 2000); the time, in which individual anabranches
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are active sediment conveyers relative to their widths or the time in which a channel
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with increasing sinuosity forms before different processes such as avulsion or cut off
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take place.
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The present knowledge about the scales of characteristic static braided river features
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is invaluable for model evaluation. The present lack of understanding in the
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relationship between spatial and temporal scales may be overcome by analysing the
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dynamic behaviour in laboratory and computational river models, which successfully
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reproduce the static scales expected from empirical studies.
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2.2. Scale invariant properties
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Various methods have been proposed to assess scale-invariant planform properties of
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braided rivers, ranging from statistical methods to state space plots in dynamical
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system theory or the theory of fractals (Nikora et al., 1991; Sapozhnikov and
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Foufoula-Georgio, 1996, 1997; Murray and Paola, 1998). Each of these methods
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reveals one or more aspects of braided rivers that act as fingerprints of a single or
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multiple characteristic mechanical processes.
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2.2.1 Sequential organisation of planform patterns ( dynamical systems approach)
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Sapazhnikov et al. (1998) used a dynamical systems approach to explore the presence
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of a sequential organization of total width in various braided rivers. They plotted
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sequences of the total channel widths per cross-section in state space plots and
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compared the plots of various rivers by dividing the state space into distinct regions
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and by assigning a probability to each region according to the number of plots that fall
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into the region. The studies suggested that braided rivers of different average
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discharge and grain sizes have a similar sequence of total widths, scaled by the
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average total width. Rosatti (2002) presented similar results for a small-scale
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laboratory flume, suggesting that laboratory rivers have a similar sequential
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organisation of standardized widths. The similarity in sequential width variations
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between different braided rivers of different scales indicates the presence of a
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characteristic distance between wide and shallow sedimentation zones and narrow,
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deep scour zones. A numerical model that incorporates sediment transport in braided
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rivers should therefore reproduce similar state space plots for standardized width
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variations. Assessment of the model according to state space plots requires the
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establishment of a measure for the similarity of state space plots of different systems
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as well as a tolerance level for the acceptance or rejection of a model. A metric to
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quantify the difference between the probability distributions corresponding to
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different rivers was defined by Moeckel and Murray (1997), but for the establishment
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of an adequate tolerance level, state space plots of a larger range of natural braided
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rivers is required.
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2.2.2 Self-affinity and dynamic scaling anisotropy (fractal methods)
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Various universal planform properties of braided rivers have been discovered using
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the theory of fractals. Sapozhnikov and Foufoula-Georgiou (1996) traced the flow
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patterns of various natural and laboratory braided rivers to separate wet from dry
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areas, and calculated mass dimensions, which are related to the number of black
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pixels (wet area) relative to the number of white pixels (dry area) within specified
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regions. This approach revealed that natural braided river channel network planform
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has fractal properties and exhibits an anisotropic scaling invariance. Smaller parts of
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the river look statistically similar to larger parts of the same river but the scaling
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differs slightly between the downstream and cross-stream direction . Moreover,
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similar fractal exponents vx and vy for the anisotropic scaling invariance were
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obtained for braided rivers of different scales, pointing towards further geometric
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similarities.
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The statistical invariance of the mechanisms generating scale invariance in the entire
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river network continues to hold for braided river islands. Sapozhnikov and Foufoula
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(1996) found anisotropic scaling in island sizes with a similar exponent for several
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braided river, which was always 10-20% lower than the scaling anisotropy of the
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rivers as a whole. Further scale invariant properties referring to island areas have been
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reported by Barzini and Ball (1993) and Rosatti (2002) for various natural and
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laboratory braided rivers, respectively.
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Caution is necessary when using fractal properties as sole evidence for model success.
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Fractal properties of channel network can only be attributed to a self-organizing
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nature of flow and sediment flux, but do not specify the particular fundamental
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processes and their degree of influence on these properties. It is therefore difficult to
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state a priori to what degree these properties hold specific for braided river systems
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and distinguish them from other self-organizing systems, such as branch structures in
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trees or dry and wet areas in coastal regions. This uncertainty has several implications
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for the evaluation of braided river models. If a braided river model does not reproduce
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these scale invariant properties, the model must lack some of the fundamental
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processes of braiding. However, since fractal properties of braided rivers cannot be
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linked directly to their causes, it is difficult to identify from this method which model
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ingredients are missing. If, on the other hand, a model does reproduce the specified
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fractal properties, it is not guaranteed that the model incorporates all of the essential
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processes contributing to a realistic braiding behaviour. The same properties may be
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generated by different processes, allowing self-organisation of flow and sediment
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flux. In addition, establishing a tolerance level for the differences between the
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proposed fractal measures is important for the application of the above proposed
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methods for the assessment of braided river models.
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2.2.3 Network link scaling
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A distinctive property of braided rivers is that characteristic measures usually spread
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widely from their averages. Some mechanical processes in braided rivers may
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therefore show their fingerprints more clearly in frequency distributions of
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characteristic measures than in their standardized average values. Ashmore (2000)
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investigated the distributions of confluence-confluence and diffluence-diffluence
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spacings standardized by the mean link length in small-scale laboratory models and
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large braided rivers and found great similarities in the shapes and value ranges of the
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distributions for various rivers (Fig. 1a). The resulting link length distributions of all
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rivers are negatively skewed and the range of link lengths covers two orders of
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magnitude.
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Like similarities in sequential width variations, similarities in the frequency
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distributions of standardized link lengths indicate the presence of a characteristic
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distance between erosion and re-deposition of sediment, although the latter lack the
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sequential information provided by the state space plots. Standard statistical tests can
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be applied to test whether different braided rivers correspond to similar frequency
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distributions of standardized link lengths.
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3. New methods to analyse braided river characteristics
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Whereas characteristic scales have been defined and analysed for both plan form and
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topographic patterns, studies of scale invariant properties in natural braided rivers
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have so far been exclusively performed for planform patterns. The reason for the lack
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of methods targeting topographic or dynamic properties lies mainly in the fact that the
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proposed methods require the coverage of a large region. Aerial photographs provide
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sufficient information about static planform properties, but analysis of the topography
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or the dynamic behaviour of braided rivers requires terrain data, which are much
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harder to acquire at present. Nevertheless, topographic properties cannot be neglected
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in the characterisation of braiding, since planform and topography are strongly linked
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by bar formation and sediment erosion and re-deposition. For example, sequential
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organisation of total channel width and frequency distributions of network links point
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towards characteristic distances between erosion and re-deposition of sediment.
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Topographic characteristics, such as scour depth and bar height, provide additional
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information about the relative quantities of transported sediment and thus add another
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dimension to the description of braiding.
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3.1. Extension of existing methods to topographic patterns
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Many of the above described methods for identifying planform characteristics could
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be extended for describing topographic properties, if sufficient topographic data are
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available. Recent progress in the development of technologies for topographic data
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acquisition makes this increasingly possible (Lane, this issue).
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The state space approach described in section 2.2.1., for example, uses information
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about the system's spatial structure. This approach could be readily extended to assess
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organization in the river topography. In this case, total width sequences could be
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replaced by sequential scour depths at channel confluences or heights of successive
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bars. The fractal methods described in section 2.2.2. use wet and dry areas in braided
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rivers to identify anisotropic scaling invariance in channel network planform patterns.
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The same techniques could prove useful for identifying scaling invariance of river
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topographies by using elevations below and above a certain reference level (e.g. water
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surface level) instead of wet and dry pixels as criteria. Further, similar distributions of
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standardized distances between successive confluences and diffluences in different
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braided rivers (section 2.2.4.) may be paralleled by similar patterns of variations in
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bed elevation along a channel, reflecting relevant characteristics in the bar-pool
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structure underlying the channel network pattern.
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3.2. Statistical scale-invariant properties of topographic patterns using transect
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data
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Not all aspects of the topography require the acquisition of a dense data set describing
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the river bed. Some useful topographic information can also be obtained from cross-
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section surveys. For example, Fig. 2 (graphs a – d) shows cross-sections of Sunwapta
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River, Canada and of physical model data (from the experiment described in Stojic et
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al., 1998). In almost all of the cross-sections, maximum scour depth below the water
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surface in a cross-section exceeds the maximum height of bars above the water
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surface. The areas of the rivers that are exposed at low discharge are also generally
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flatter than the areas that remain under water.
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Thus, the elevation median generally exceeds the elevation mean and the standardized
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frequency distribution of deviations from the elevation median is skewed (Fig. 1b).
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The deviations from the median in Fig. 1b are standardized by the standard deviation
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σ = [1/n × Σ ni=1(zi – zm)2] 0.5, where zm is the elevation median for the cross-section
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and zi are the measured elevations.
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Both prototype and model distributions have similar shapes and ranges of scaled
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deviations from the median. The two-sample Kolmogorov-Smirnov test with a 95%
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confidence level identifies the frequency distributions of the deviations from the
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median for the Sunwapta and the laboratory river as samples from the same
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population. The strong similarity between the distributions is surprising, particularly
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as differences in the distributions are expected due to the different data acquisition
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techniques (point survey versus digital photogrammetry) and flow properties
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(discharge was held constant in the laboratory model). The boundaries of the transect
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elevations in the Sunwapta river frequently consist of high elevation values
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corresponding to steep banks and artificial rip-rapped embankments (Fig. 2a and b),
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which may account for the higher occurrence of deviations greater than one standard
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deviation from the median in the positive direction.
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An estimate for the degree of structure in the topography can be derived from a simple
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approach based on the frequency of elevations crossing a reference level, for example
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the median elevation, per river width (cross-section). In this example, to allow
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comparison between rivers of very different scale, cross-sections were divided into
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the same number of subintervals and the number of relative sign changes per cross-
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section was defined as the ratio between the number of actual sign changes and the
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number of subintervals. Averaged over many cross-sections and various sampling
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times, the Sunwapta River has 0.13 relative sign changes per cross-section. For the
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laboratory river in fully braided state, the corresponding averages are 0.12 relative
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sign changes per cross-section. The temporal variations about these averages
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exhibited a random behaviour and were about 14% for the Sunwapta River and about
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10% for the laboratory river.
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The prevalence of relatively wide and flat areas above the water surface and relatively
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narrow, deep areas below the water surface can be captured by distributions of lateral
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bed slopes. Using the elevation median as a reference level instead of the water
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surface level, which is often unknown and often exhibits strong variations, lateral
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slope distributions below and above the elevation median have distinct shapes. The
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frequency distribution curves of lateral slopes corresponding to areas with elevations
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less than the elevation median are significantly wider than those corresponding to the
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areas with elevations above the median (Fig. 1c, d), reflecting the observed shape
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differences between exposed and submerged areas. Further, the ranges of lateral
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slopes corresponding to elevations below the median elevation are similar for both
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rivers. For elevations above the median, the variation of lateral slope values is
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however significantly higher in the Sunwapta River.
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The two-sampled Kolmogorov-Smirnov test with a 95% confidence level identifies
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the distributions of lateral slopes below the elevation median for the Sunwapta and
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laboratory river as samples from the same population, but the distributions
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corresponding to elevations above the elevation median as samples from different
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populations.
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Mechanical properties such as shear strength of the sand or gravel could contribute to
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the observed discrepancy in the lateral slope distributions. In this case the distribution
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of lateral slopes could not be classified as scale invariant properties of braided rivers
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of different scales. However, since the differences are stronger for elevations above
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the median, it is likely that the observed discrepancy results from the temporal
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variation in discharge in the natural river in contrast to the constant discharge in the
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flume. In the annual summer-melt, water covers a high percentage of the usually
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exposed area and transforms the otherwise inactive bars to erosion and deposition
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sites with steeper lateral slopes.
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The overall similarities between various frequency distributions derived from
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topographic transect data of different rivers point towards the presence of scale
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invariant properties in the river topographies. The observed discrepancies between the
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distributions of lateral slopes suggest, however, that some distributions are more
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sensitive to variations in hydraulic conditions or material properties than others.
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Topographic data from a greater variety of braided rivers of different scales are
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needed to test the robustness and sensitivity of the introduced methods.
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[INSERT FIGURES 1 AND 2 HERE]
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4. Using characteristic scales and scale invariant properties to evaluate the MP
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model for braided rivers
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Methods targeting only plan form properties of braided rivers on a network-scale are
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not sufficient for assessing whether a model produces a realistic braided behaviour,
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but must be augmented by methods that relate to various characteristic features in
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braided rivers including, in particular, topographic characteristics. The cellular model
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of Murray and Paola (1994) serves to demonstrate the gains of model evaluation
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according to multiple criteria. This model lends itself to the present study for various
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reasons:
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It was designed to model the dynamic behaviour of a large braided network using
equations that describe the interaction between flow and topography.
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The model is based on simple concepts and is therefore not designed to simulate
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the behaviour of a particular braided river. Its purpose is not prediction of future
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behaviour of a particular river, but rather aims to enhance the understanding of the
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fundamental processes taking place in braided rivers. Model evaluation is
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therefore challenging as it cannot be restricted to a direct comparison between the
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modelled and a particular natural river, but must comprise tools that refer to
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properties that are characteristic for braided rivers in general. In fact, the model
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has provoked the development of various quantitative methods for the assessment
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of modelled braided river planforms (Murray and Paola, 1998, Sapozhnikov et
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al., 1998).
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The model has attracted much attention amongst braided river researchers and
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stimulated additional development and application because it reproduces many
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characteristic aspects of braided rivers from simple rules (Coulthard, 1999,
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Thomas et al., 2002, Doeschl et al. 2003, Murray and Paola, 2003).
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Although extensive model evaluation has been undertaken (Murray and Paola,
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1994, 1997, 1998, Sapozhnikov et al., 1998), the analysis of its performance is
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incomplete because it has concentrated only on planform characteristics at the
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scale of the braided network. Visual inspection of the topographic patterns
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suggests strong discrepancies between the topographies of the modelled and
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natural braided rivers. There remains uncertainty about the extent to which the
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model reproduces braided river morpho-dynamics. Its validity with respect to
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producing both, realistic planform and topographic characteristics along with
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realistic dynamic behaviour of these properties has not been tested.
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The subsequent evaluation of the MP model is preceded by a brief summary of the
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model concepts and previous results.
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4.1. The MP model and previous results
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The MP model uses the concept of cellular automata theory to generate a continuously
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changing braided river network from random initial conditions.
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It operates on a 2D lattice of square cells, along which water and sediment are routed
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into the downstream direction and lateral direction (the latter refers to sediment
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transport only) according to the equations
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11
Q
= Q0 Sn / Σ Sjn
12
Qs
= K ( Q S + ε Σ(Qj Sj))m
13
Qs,l = Kl Sl Qsout,
if S >0
(1)
(2)
(3)
14
15
where Q is the fraction of local discharge delivered to a neighbouring cell from the
16
total discharge Q0 in the distributing cell, S is the local bed slope, Qs and Qs,l are the
17
amounts of sediment transferred between the downstream and lateral cells,
18
respectively. The subscript j refers to the three downstream neighbours of a
19
distributing cell, and n, m, K, ε and Kl are model parameters. The original model
20
included some variations of the downstream sediment transport rule, but for the
21
present study, rule (2) was used. Change in cell elevation as a result of sediment
22
movement obeys the mass balance law. Equations (1) to (3) are representations of
23
various physical laws, including the law of conservation of mass for water flux and
24
Exner’s equation for mass balance in the sediment flux as well as crude
25
representations of the bedload formula of Engelund and Hansen (Engelund and
Doeschl et al.
20
1
Hansen, 1967; Murray and Paola, 1997) and the lateral sediment transport formula by
2
Parker, (Parker, 1984; Murray and Paola, 1997). A detailed description of the model is
3
provided in (Murray and Paola, 1997).
4
5
The model was not designed for prediction purposes, but rather aimed to enhance the
6
understanding of braiding by finding the conditions required to generate realistic
7
braided patterns that retain the characteristic instability of braided rivers. In Murray
8
and Paola’s computational experiments, the model started from a scale-free initial
9
topography of constant downhill slope with superimposed random perturbations.
10
Model parameters were not calibrated to simulate a particular physical setup, but were
11
chosen freely, on the basis of how realistic the generated patterns looked. As a
12
consequence, the generated patterns have an intrinsic scale, which is difficult to
13
compare with that of natural braided rivers. Although Murray and Paola proposed a
14
method to relate a time scale with the scaling constant K in the sediment transport rule
15
(2) (Murray and Paola, 1997), this setup did not lend itself to a direct comparison of
16
the characteristic scales of the generated modelled rivers with those of natural rivers.
17
Model evaluation concentrated therefore on the assessment of scale invariant
18
properties (Murray and Paola, 1996, 1997; Sapazhnikov et al., 1998).
19
20
State space plots revealed strong similarities between the sequences of total cross-
21
sectional widths of the modelled rivers and several natural rivers. Although the
22
probability distributions associated with the width sequences of the modelled and
23
various natural rivers were statistically identified as distributions from different
24
populations, the corresponding transportation distances were relatively small
25
(Sapozhnikov et al., 1998).
Doeschl et al.
21
1
Like natural braided rivers, flow and sediment flux in the modelled rivers were found
2
to exhibit a self-organizing nature, which is expressed by a self-affine behaviour of
3
the planform patterns (Sapozhnikov et al., 1998). Moreover, the agreement between
4
spatial scaling characteristics, such as the ranges of the fractal exponents, was
5
generally high between natural, laboratory and modelled rivers. The scaling
6
anisotropy of islands as opposed to the entire river network was found about 10-20%
7
percent lower in natural and the modelled braided rivers.
8
9
Reproduction of these scale invariant properties in the modelled rivers could be
10
considered as an indication that the model correctly represents the mechanisms that
11
control the planform structure of braided rivers. However, subtle differences between
12
the modelled and natural rivers, which were also observed, may point to some missing
13
ingredients in the MP model. For example, statistical tests indicated significant
14
differences between the distribution of transportation distances between the modelled
15
and natural braided rivers. Subtle discrepancies were also found in the conditions for
16
sudden width variations. In natural braided rivers, these occurred predominantly in
17
wide sections, while they occurred equally in narrow and wide sections in the
18
modelled rivers (Sapozhnikov et al., 1998). Further, modelled rivers were found to
19
exhibit an on average higher scaling anisotropy with higher variance than natural
20
rivers, which reflects the presence of fewer channels and the absence of a long-scale
21
sinuosity in the modelled rivers (Sapozhnikov et al., 1998).
22
Without estimates of error bars associated with the variations of these scale invariant
23
properties in natural braided rivers or a thorough understanding of the relationship
24
between the observed symptoms and the underlying controls, it is however difficult to
25
determine, to what extent the observed discrepancies point to actual imperfection in
Doeschl et al.
22
1
the model equations. The answer to this problem requires therefore the establishment
2
of a tolerance level within which a model is accepted or the implementation of
3
additional methods for describing planform characteristics.
4
5
The MP model produces also detailed information about the topographies of the
6
modelled rivers, which, after visual inspection raise some doubt about their
7
resemblance to those of natural braided rivers. This is difficult to describe but while
8
real braided rivers show clear channels and bars with gradual transitions between
9
areas of low and high elevation (e.g. Westaway et al., 2003), the topography
10
generated by MP model appears less organized with abrupt transitions in elevation.
11
However, except for the observed association between the formation of scour holes
12
and bars and flow contraction and expansion, respectively (Murray and Paola, 1997),
13
no detailed validation of the modelled topographies have however been performed.
14
15
4.2. Assessment of the characteristic scales of the modelled rivers
16
On the basis that in natural braided rivers the underlying mechanical processes are
17
expressed in the rivers’ characteristic scales and their scale invariant properties, the
18
subsequent model evaluation is also divided into these two parts. This section deals
19
with the scales of planform and topographic quantities produced by the MP model.
20
21
The dimension of the computational grid representing the braid plain, the introduced
22
water and sediment fluxes, and the values of the model input parameters, control the
23
structure and scales of the characteristic features of the modelled rivers. Under the
24
assumption that the model incorporates the main mechanisms that lead to a realistic
25
braided behaviour, it is hypothesized that the model would also reproduce the
Doeschl et al.
23
1
characteristic scales of a physical braided river, if the necessary information for initial
2
values and parameter estimates corresponding to those of the natural river were
3
available. To test this hypothesis, the MP model was implemented to simulate the
4
initial setup of a laboratory flume experiment, for which the required information was
5
available.
6
7
This approach provides the possibility of assigning a spatial or temporal scale to the
8
modelled river networks, and enables a direct comparison between the characteristic
9
spatial and temporal scales of the modelled and the physical river planforms and
10
topographies. Since many empirical equations that relate hydraulic and geometric
11
conditions to scales of planform and topographic patterns, have not been tested for
12
small-scale laboratory of braided rivers, it cannot be assumed a priori that the flume
13
model reproduces the expected spatial and temporal scales. For this reason the scales
14
of both, the laboratory and the computationally modelled rivers, are compared with
15
those derived from natural braided rivers under equivalent external forcing.
16
17
The flume experiment
18
The laboratory river used for this study approximately represented a 1.40 generic
19
scale hydraulic model of a gravel-bed braided stream. The flume experiment consisted
20
of time periods (typically 10 – 15 minutes) during which water and sediment were
21
introduced at a constant rate at the upstream end and the river morphology evolved.
22
At the end of each period water was drained from the flume and digital photographs
23
were taken of the bed, from which a sequential set of digital elevation models of the
24
flume-bed was derived (Stojic et al, 1998; Fig. 3). In the digital elevation models, the
25
river bed is divided into a grid of cells with (x,y,z) coordinates, which has the same
Doeschl et al.
24
1
structure as the computational lattice in the MP model. Measures describing the flume
2
dimensions and the physical setup are listed in Table 1.
3
4
The experimental river evolved from an initially straight channel to meandering and
5
finally to braided patterns. The evolution towards a fully braided stage took
6
approximately 3 hours. Once the braided stage was developed, a statistical stability
7
was reached, which is characterized by small temporal fluctuations but overall
8
constant values of characteristic measures such as the braiding index, average channel
9
widths and average scour depth / bar height. Despite this robustness in the statistical
10
measures, the river constantly reconfigured itself and exhibited a vast range of
11
characteristic dynamical processes, such as avulsion, channel migrations, chute cut-
12
off, etc. The consequences of these dynamic processes on the bed topography are
13
shown in Fig. 3.
14
15
[INSERT FIGURE 3 HERE]
16
17
Adoption of the setup of the flume experiment
18
The dimension of the computational lattice, the initial values for the model variables
19
and the model input parameters in the MP model were calculated from the dimension
20
of the flume, the median grain size and the hydraulic conditions for the flume
21
experiment (Table 1).
22
23
The flume’s initial topography was represented by a smooth surface of the flume’s
24
constant downhill slope and superimposed white noise. The maximal amplitudes of
25
the white noise perturbations were chosen to be 10 times the valley slope. However,
Doeschl et al.
25
1
computational experiments with different amplitudes of perturbation showed that the
2
amplitudes have little impact on the planform and topographic scales of the fully
3
developed modelled braided rivers. Dimension and resolution of the computational
4
lattice were chosen equal to the raster structure of the digital elevation models. The
5
time step corresponding to one iteration of the computational model was determined
6
by the initial volumetric discharge and the grid spacing. This setup allowed a 1:1
7
comparison between the spatial and temporal scales of the MP modelled rivers and the
8
laboratory river.
9
10
Due to their heuristic origin, calibration of the five model parameters n, m, K, ε and
11
Kl from the physical experiment was not straight-forward. Following Murray and
12
Paola’s outline (1997), the parameters K and m could be derived from Engelund and
13
Hansen’s sediment transport formula (Engelund and Hansen, 1967). Using uniform
14
flow formulae, the latter formula can be rewritten in form of equation (2) with ε = 0,
15
m=5/3 and K = 0.4 × [5.66 g 1/3 f 1/6 D50 (s-1)]-1, where f is the Darcy Weisbach
16
friction coefficient, D50 is the average grain size and s is the specific gravity.
17
Substituting appropriate values into the equation for K leads to K = 2.4 × 10-5.
18
Similarly, the lateral sediment transport equation (3) could be considered as a crude
19
representation of Parker’s lateral sediment transport formula (Parker, 1984), which
20
yielded Kl = 0.875, after substituting appropriate values of the average water depth in
21
the flume experiment, grain diameter and bed slope and an estimate of the critical
22
shear stress on the level bed (Doeschl, 2002). The parameters n and ε were
23
unconstraint and were chosen to obtain optimal results.
24
Doeschl et al.
26
1
Obviously, the modelled river is not expected to exhibit a similar evolution as the
2
laboratory river. The question is, whether the MP model represents the physics of the
3
laboratory river sufficiently to generate braided patterns and topographic features of
4
similar scales as those in the physical braided river, and whether the modelled rivers
5
exhibit a similarly rich dynamic behaviour with similar intrinsic time scales as the
6
laboratory river.
7
8
Comparison of the evolution of the laboratory and the modelled river
9
The above parameter estimates together with n>1 and 0.3  ε  0.5 led to braiding, but
10
with unrealistic channel structures and poor structural changes over time. More
11
realistic braided patterns, which constantly reconfigured, could however be generated
12
by increasing the lateral scaling factor to Kl = 20 and the sediment transport exponent
13
m to 2.5.
14
15
In contrast to the laboratory river, which evolved from straight to meandering and
16
finally braided patterns in approximately 3 hours, the MP modelled river has a large
17
number of frequently intertwining channels during the first few hours flume time with
18
small and rapidly altering wet and dry areas (Fig. 4a, b). Eventually several narrow
19
channels combine to fewer wider channels and the wet and dry areas increase in size.
20
Only after approximately 30 hours flume time is a statistically stable state reached, at
21
which the river constantly reconfigures without changing the average channel width
22
or the average number of channels per cross-section (Fig. 4c-f). However, in contrast
23
to the statistical equilibrium in the planform structure, bars and pools keep increasing
24
in size in the MP modelled rivers (Fig. 4b, d, f and h). This increasing trend in
25
topographic features eventually affects the statistical balance in the planform pattern.
Doeschl et al.
27
1
After many hours flume time, the braiding index gradually decreases until the MP
2
modelled river is no longer braided.
3
4
[INSERT FIGURE 4 HERE]
5
6
Comparison of the characteristic scales of the laboratory and the modelled river
7
Characteristic scales help to quantify the substantial differences between the
8
laboratory and the MP modelled river as suspected just from visual inspection.
9
It is obvious from the temporal sequences of digital elevation models (Fig. 3 and 4),
10
that the MP modelled river exhibits different spatial and temporal scales than the
11
laboratory river. Table 2 provides time averaged values of some characteristic scales
12
for both rivers in their statistically stable braided stages, as well as some estimates
13
derived from empirical studies on natural braided rivers. The measures describing
14
morphological properties in the MP modelled rivers, which are marked with
15
subscripts + and -, have a strictly increasing (+) or decreasing (-) tendency with
16
respect to time. Hence, the averages shown in Table 2 are biased by the times for
17
which the values reach their maximum.
18
19
For most of the measures, the agreement between the measured values and the values
20
predicted by empirical studies for the laboratory river is strong (Table 2). The
21
measured total width exceeds the predicted width by a factor greater than two, but lies
22
still within the 95% confidence limits. In contrast, the discrepancy between the
23
characteristic scales of the MP modelled river and those of the laboratory river is very
24
high. Compared to the laboratory river, the MP modelled river has a larger number of
25
narrower channels, which also split and join more frequently. Consequently, the
Doeschl et al.
28
1
modelled river has a higher frequency of pools and bars than the flume. Even after
2
130 hours simulated flume time, the continuously increasing magnitudes and areas of
3
bars and pools are on average an order of magnitude smaller than those in the flume
4
experiment. Similarly, the total sinuosity in the laboratory and the MP modelled river
5
differs by an order of magnitude.
6
The time scale associated with the evolution of braiding is considerably longer in the
7
MP modelled river. Development towards the statistical equilibrium in the planform
8
structure differs approximately by a factor 10.
9
10
Influence of the raster structure of the computational grid on the spatial scales
11
To determine the impact of the dimension of the computational grid on the scales of
12
the MP modelled river, the model outputs associated with the same parameter
13
combinations, but various grid dimensions were compared. Surprisingly, the spatial
14
scales of the model predictions, when measured in cell units, show little dependence
15
on the grid structure. Figure 5 shows that channel width and the spacings between
16
successive confluences and diffluences vary little between different grid dimensions.
17
For cell lengths ranging from 1/2 times to 4 times the chosen cell lengths of the raster
18
structure for laboratory flume, the average channel width varies between 4.18 and
19
4.32 cells; a drop in the channel width occurs only for extremely coarse grid
20
structures. The average spacing between adjacent nodes varies between 22.5 and 28
21
cells.
22
23
The invariance of the average channel width and nodal spacing in terms of cell units
24
has several implications. First, similar planform scales as those observed in the
25
laboratory river could be obtained by the MP model through a change of the cell
Doeschl et al.
29
1
dimensions in the computational lattice. In this case, model assessment using
2
exclusively planform patterns would lead to misleading results, since only the scales
3
of the topographic features (e.g. average scour depth and bar height) point out
4
substantial differences between the MP modelled and the laboratory river. Second, the
5
observed invariance suggests that the model possesses an intrinsic scale on which its
6
laws operate. This would imply that the MP model is not able to represent rivers of
7
different scales, but generates a river of the particular scale on which the model
8
equations operate.
9
10
[INSERT FIGURE 5 HERE]
11
12
Implications
13
The above results show that a proper representation of the setup of the physical
14
experiments in the MP model does not lead to similar spatial and temporal scales for
15
characteristic features as those observed in the physical experiment. Whereas the
16
spatial scales associated with the laboratory river generally agree with the scales
17
predicted by empirical studies, the MP modelled river produces different spatial scales
18
in both planform and topographic patterns. The temporal scales associated with the
19
evolution towards a statistical equilibrium state and structural morphological changes
20
disagree between the laboratory and MP modelled river.
21
The model’s failures to generate a statistically stable topography and to produce
22
characteristic patterns of the expected scales leads to the inference that the model does
23
not represent sufficiently all physical components that are necessary to produce a
24
realistic braided behaviour. Two model variables, i.e. water and sediment flux, and
25
local bed gradients as the driving forces for planform and topographic evolution may
Doeschl et al.
30
1
not suffice for this purpose and may be responsible for the observed intrinsic scales of
2
the generated patterns.
3
The suspicions arising from the present studies are supported by results obtained in
4
complementary studies, in which the MP model was applied to braided topographies
5
of natural and laboratory rivers (Thomas and Nicolas, 2002; Doeschl et al., 2003). It
6
was found that the MP model, without modifications, could not predict a realistic
7
evolution of the corresponding braided rivers. However, in both computational
8
experiments, model results improved considerably through the incorporation of
9
additional model variables and more detailed representation of the physical laws.
10
11
4.3. Assessment of scale invariant properties of the modelled rivers
12
13
4.3.1. Planform characteristics
14
The previous assessment of planform characteristics (Murray and Paola, 1996;
15
Sapozhnikov et al., 1998) can be augmented by the assessment of the distribution of
16
standardized link lengths from the modelled channel network.
17
Figure 6a shows that the cumulative frequency distribution of standardized link-
18
lengths of the modelled river, calculated over various time intervals during the fully
19
braided stage, shares the main characteristics such as skewness and link-length-range
20
with those of various natural and laboratory rivers. The Kolmogorov-Smirnov test,
21
with a 95% confidence level, does not identify the distributions of the modelled river
22
and any of the physical rivers as significantly different. The overall similarity in the
23
link-length distributions supports thus the results of previous studies, i.e. that the
24
modelled planform patterns resemble those of real braided rivers, when transformed
25
to scale-invariant quantities.
Doeschl et al.
31
1
2
4.3.2. Topographic characteristics
3
Transect graphs and statistical measures
4
The transect graphs of the topographies of the Sunwapta River and the laboratory
5
river show that areas above the elevation median are predominantly wide and flat,
6
whereas areas below the median are more narrow and sharply peaked. Areas above
7
the median with relatively steep lateral slopes occur seldom in the flume cross-
8
sections and primarily at the boundaries of the braid plain in the Sunwapta River.
9
Transects of the MP modelled topographies do not share these properties as many
10
cross-sections are characterized by the coexistence of sharp peaks above and below
11
the elevation median (Fig. 2e, f). Consequently, the elevation mean exceeds more
12
frequently the elevation median in the modelled than in the physical rivers. Under the
13
given constant flow conditions, the appearance of sharp peaks above the elevation
14
median in the braid plain centre cannot be attributed to rising and falling flow stages,
15
but is more likely a result of the cellular nature of the model. The distribution of flow
16
and sediment according to local slopes between adjacent cells allows rapid direction
17
changes in the flow and the occurrence of abrupt deposition of large quantities of
18
sediment as an immediate consequence of this shift in direction.
19
20
Degree of structure in the modelled topography
21
The temporal trend in the topographic structure of the modelled river, which became
22
evident from the visual inspection of the modelled topographies at sequential times, is
23
captured by the relative frequencies of sign changes in the deviations of cell
24
elevations from the cross-sectional median. During fully braided states, the cross-
25
sectional averages of the MP modelled river decreased from 0.26 relative sign
Doeschl et al.
32
1
changes at 30 hours flume time, to 0.18 relative sign changes at 160 hours flume time.
2
In contrast, the Sunwapta River and the laboratory river exhibited on average 0.13 and
3
0.12 relative sign changes, respectively, with small, random temporal fluctuations.
4
The decreasing trend of relative sign changes in the modelled topography thus
5
contrasts with the statistically stable evolution of the topographies in physical braided
6
river systems. Little inference can be made about the higher frequency of elevations
7
crossing the median elevation in the MP modelled river compared to the physical
8
rivers, as it may be a genuine sign for a higher degree of randomness in the modelled
9
topographies or could just be a relict of the structure of the initial topography, which
10
was represented by random perturbations of a flat topography.
11
12
Frequency distributions of deviations from the elevation median
13
Figure 6b shows the frequency distributions for the standardized deviations from the
14
elevation median for the Sunwapta River, the laboratory and the MP modelled rivers.
15
The distributions of all three rivers have similar shapes and cover a similar spectrum
16
of value ranges. The Kolmogorov-Smirnov test with a 95% confidence limit does not
17
detect a significant difference between the MP modelled and the physical rivers. In
18
particular, the topographic trend in the MP modelled river is not captured by these
19
frequency distributions; frequency distributions associated with 30 hours flume time
20
and 140 hours flume time are not significantly different according to the Kolmogorov-
21
Smirnov test with a 95% confidence level.
22
23
Frequency distributions of lateral bed slopes
24
Although quite different in their scales, the Sunwapta River and the laboratory river
25
have a similar value range of lateral bed slopes (Fig. 1c,d). The lateral bed slopes
Doeschl et al.
33
1
of the MP modelled river, however, are generally an order of magnitude lower (Fig.
2
6d). This discrepancy is associated with the smaller magnitudes of bed elevations in
3
the MP modelled river compared to the laboratory river (Section 4.2) and provides
4
further evidence that the topographies generated by the MP model differ from those of
5
physical braided rivers in essential aspects.
6
7
However, as in the physical rivers, the distribution of lateral slopes for areas below
8
the elevation median is wider than for areas above the median in the modelled river
9
(Fig. 1c, d and Fig. 6c, d), reflecting the higher frequency of relatively steep lateral
10
slopes in submerged than exposed areas. But whereas in the Sunwapta and the
11
laboratory river, lateral slopes in the extreme 10 percentiles of the lateral slope ranges
12
for elevations above the median occur extremely rarely, the lateral slopes
13
corresponding to areas above the elevation median in the MP modelled rivers assume
14
extreme values with a similar frequency as those corresponding to elevations below
15
the elevation median. The lateral slope distributions thus quantify the observed
16
coexistence of sharp peaks in areas below and above the elevation median in the MP
17
modelled rivers.
18
19
[INSERT FIGURE 6 HERE]
20
21
4.3.3. Implications
22
Combination of the results of all methods utilized for the assessment of the scale
23
invariant planform and topographic characteristics of the MP modelled rivers, yields
24
an ambiguous picture of the model performance. Whereas all previously and newly
25
applied scale-invariant methods point to strong similarities of planform properties on
Doeschl et al.
34
1
a network scale between the MP modelled and various physical rivers, considerable
2
differences could be found in the topography of these rivers.
3
4
Different methods were required to capture relevant similarities and differences
5
between the topographies generated by the MP model and those of physical braided
6
rivers. The frequency distributions of deviations from the elevation median and the
7
distributions of lateral bed slopes represent the fact that in the modelled, as in natural
8
rivers, the maximal scour depth generally exceeds the maximum bar height and
9
relatively steep lateral slopes occur predominantly in submerged areas. The frequency
10
distributions of the deviations from the elevation median could, however, not capture
11
any differences between the MP modelled and natural rivers. Structural differences
12
are instead represented in both, the distributions of lateral bed slopes and in the
13
frequencies of sign changes for crossing the elevation median. The observed trend in
14
the topographic structure is only represented in the latter method.
15
16
Although the differences in the topography do apparently not affect planform
17
characteristics on a network scale, they certainly affect the planform characteristics on
18
the pool/bar scale, as can be seen from visual inspection of the model outputs.
19
Quantitative analysis of planform properties should therefore be extended from
20
network-scale to smaller-scale properties.
21
22
Discussion
23
Braided river modelling encompasses a variety of approaches (for a review of some
24
existing approaches see Jagers, 2003). According to Murray (2003), simulation
25
models, which aim to reproduce or predict the behaviour of a particular river form one
Doeschl et al.
35
1
end of the continuous spectrum of modelling approaches. Evaluation of the output of
2
simulation models is based on direct comparison between the modelled river and the
3
past behaviour of the simulated natural river. On the other end of the spectrum stand
4
exploratory models, which target to explain poorly understood phenomena such as the
5
interaction between water and sediment flux and the river morphology, using often
6
more relations based on conception than established physics. The MP model clearly
7
falls into this category. Evaluation of these types of models can be challenging,
8
because it needs to be decided whether the model output reproduces a morphology
9
and dynamics that is realistic for a braided river in general.
10
11
However different the modelling approaches, central for the evaluation of each
12
braided river model is a proper quantification of braided river characteristics that
13
allow a modeller to assess the extent to which the model output reproduces the
14
phenomena. Although the quantitative description of braided river phenomena is at
15
present dominated by static planform characteristics, topographic characteristics and
16
the dynamic behaviour of the morphology are equally important for describing
17
braiding. Especially for models that incorporate rules representing the interaction of
18
flow, sediment flux and planform- and topographical evolution, such as the MP
19
model, assessment according to criteria considering only one of the braided river
20
constituents, can be misleading.
21
22
We are not the first to argue that multiple quantities with coupled behaviour should be
23
used to test models. Furbish (2003) warns that focussing on one quantity is unlikely to
24
provide a sufficient discriminative basis and insists that a model should
25
simultaneously predict at least two unambiguously coupled quantities. He further
Doeschl et al.
36
1
concludes that if the model cannot correctly reproduce the salient features of both
2
quantities together, some part of the model must be wrong. The potential benefits of
3
imposing additional criteria on model predictions and thus of providing more
4
demanding tests are a fuller understanding of a model’s capacities and limits and
5
eventually the creation of a superior model.
6
7
On this basis, we combined existing with newly introduced quantities that prove fit
8
for model evaluation and used them to test the cellular MP model. The quantities
9
divide into characteristic spatial and temporal scales, which are the river’s response to
10
the scales of the external forcing, and into statistical scale-invariant properties that
11
quantify common phenomena in all braided rivers of different scales and which
12
emerge from fundamental processes inherent in all braided rivers.
13
14
In order to assess the MP modelled rivers according to characteristic scales, i.e.
15
according to criteria that are usually only applied to simulation models, initial
16
conditions and parameter values in the model were set (as far as possible) such that
17
the model simulates the setup of a physical laboratory river, for which detailed
18
information of spatial and temporal scales exists. This experiment revealed that, in
19
contrast to the laboratory braided river, the planform and topographic features of the
20
MP modelled river do not match the scales predicted from empirical studies with
21
various natural braided rivers and thus, that the MP modelled river does not respond
22
correctly to external forcing. These results are not surprising, since the MP model was
23
never designed to simulate a specific natural or laboratory river. But these results
24
combined with the results from independent studies (Coulthard et al., 2000; Thomas
25
and Nicholas, 2002; Doeschl et al., 2003), which have shown that the model’s
Doeschl et al.
37
1
performance for predicting the evolution of a particular system improves through the
2
addition of more variables and physical laws, suggest that incorporation of elements
3
of fundamental physics is necessary, if a model is expected to generate features of the
4
appropriate scales.
5
6
Accepting that the MP model generates rivers of different scales than natural or
7
laboratory rivers under equivalent hydraulic and geometric conditions, we then
8
proceeded to test whether the MP model reproduces characteristic static planform and
9
topographic properties that are common for all braided rivers of different scales.
10
Previous and novel methods for testing planform characteristics on the network scale
11
lead to the inference that the channel network structure of the MP modelled rivers
12
resembles strongly the network structure of natural and laboratory braided rivers.
13
Novel methods, targeting topographic characteristics, point however towards
14
significant discrepancies between the MP modelled topographies and those of natural
15
and laboratory braided rivers. The most important lesson to be learnt from this
16
analysis is that some statistical method discern the differences between modelled and
17
real braiding better than others and that a variety of methods are needed to provide a
18
full understanding of the capacity of the model to reproduce ‘real braiding’. Due to
19
the strong link between topography and planform, the observed topographic
20
discrepancies put the previously observed similarities in the network planform
21
characteristics into question and suggest that more discriminating methods or levels of
22
acceptance may be needed to discern unrealistic properties of the channel network
23
structure. In addition, methods that target planform characteristics of a scale smaller
24
than the entire network may be more suitable to point towards missing model
25
components.
Doeschl et al.
38
1
2
Although pushed beyond its initial purpose as pure exploratory model, the
3
implementation of the MP model to predict the evolution of a laboratory braided river
4
provides many benefits apart from the fact that characteristic phenomena can be
5
quantified by their scales. The direct comparison between the static and dynamic
6
behaviour of the modelled and the laboratory river offers useful insights into the
7
aspects of braiding that should be captured by quantitative characterisation tools. The
8
characteristic scales and scale invariant static properties used for the model
9
assessment in this study cover only a small number of braiding characteristics. Static
10
characteristics that are not encompassed by the present methods, but which become
11
evident through the visual inspection of modelled and real braided rivers, include the
12
location, shape and orientation of scour holes and deposition zones in relation to
13
channel confluences and diffluences, the fraction of the channels that are actively
14
eroding or depositing sediment or measures describing channel curvatures relative to
15
the channel width. Characteristics describing braided river dynamics encompass the
16
spatial and temporal sequence of events like channel migration, avulsion or chute cut-
17
off, the rate of bar migration relative to bend migration or the range of magnitude and
18
shape of erosion and deposition zones relative to the average sediment transport rate.
19
20
Prerequisite for the quantification of the proposed and other characteristics is the
21
acquisition of dense flow and topographic data of natural and laboratory braided
22
rivers over sufficiently long time intervals. The present advance in the development of
23
reliable measuring technology for the acquisition of digital terrain data is therefore
24
vital for the progress in model evaluation and therefore also for the progress in model
25
development. The acquisition of topographic data would enable us to confirm or reject
Doeschl et al.
39
1
the preliminary results of this study based on a sparse set of topographic data and
2
would also enable us to apply the statistical, fractal and dynamical systems methods
3
currently used to examine planform patterns, to topographical features. Moreover,
4
analysis of topographic and planform data may provoke the development of novel
5
methods, which capture characteristics that have so far only been described
6
qualitatively.
7
8
The obvious shortcoming of this present study is its focus on static properties of
9
braided rivers. At present, the relationship between spatial and temporal
10
characteristics and scales is poorly understood. Sapozhnikov and Foufoula-Georgiou
11
(1997) used fractal methods to identify some universal dynamic aspects of braided
12
river planforms. They detected that braided rivers exhibit a dynamic scaling
13
invariance, meaning that a smaller part of a river evolves statistically identically to a
14
larger part, provided that the time is rescaled by a factor which depends only on the
15
spatial scale of the two parts. More accurately, a dynamic scaling anisotropy implies
16
that for different square regions of size lengths L1 and L2 respectively, there exists
17
an exponent z such that T2 / T1 = (L2 / L1)z , where T1 and T2 are characteristic
18
times associated with the evolution within areas of size lengths L1 and L2.
19
Experimental studies yield an estimate of 0.5 for the exponent z. Whereas the
20
dynamic scaling exponent z expresses the relationship between spatial and temporal
21
scales of different sections of the same braided river, dimensional analysis expresses
22
the same relationship between the spatial and temporal scales of different braided
23
rivers with the same Froude-number. This property is the basis for laboratory small-
24
scale models of braided rivers. Apart from its importance as an additional universal
25
property in braided rivers, dynamic scaling anisotropy is at present the only empirical
Doeschl et al.
40
1
evidence for a close relationship between spatial and temporal scales in rivers.
2
Theoretical stability analysis (Tubino, 1991) provides further evidence for a strong
3
relationship between spatial and temporal scales.
4
It is likely that characteristic patterns of morphological changes in braided rivers
5
would be revealed using the same fractal methods utilized for describing self-affinity
6
in the planform structure (section 2.2.2). Common patterns of erosion or deposition in
7
natural braided rivers could be assessed by using areas of erosion or deposition during
8
fixed time periods instead of wet and dry areas or island shapes. Further progress in
9
the assessment of characteristics in braided river dynamics thus also depends on the
10
future availability of digital terrain data of natural and laboratory rivers over a
11
prolonged period of time.
12
13
14
15
16
17
18
19
Doeschl et al.
41
1
References:
2
3
Ashmore, P. and Church, M. (1995) Sediment transport and river morphology: a
4
paradigm for study. In: Gravel-bed rivers in the Environment. (Ed. Klingeman, P.C.,
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Beschta, R.L., Komar, P.D. and Bradley, J.B.) Water Resources Publications LLc,
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8
Ashmore, P. (1991) Channel morphology and bed load pulses in braided, gravel-bed
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Ashmore, P. (2000) Braiding phenomena: statics and kinetics. Gravel-bed rivers 5.
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Bridge, J.S. (1993) The interaction between channel geometry, water flow, sediment
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Coulthard, T.J., Kirkby, M.J. and Macklin, G.M. (2000) Modellig geormorphic
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Doeschl, A. (2002) Assessing braided rivers with a cellular model. PhD thesis.
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Engelund, F. and Hansen, E. (1967) A monograph of sediment transport in alluvial
6
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Furbish, D.J. (2003). Using the dynamically coupled behaviour of land-surface
9
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Predictions in Geomorphology (Ed. Wilcock, P. R. and Iverson, R.M.) Geophysical
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Hoey, T.B. (1992). Temporal variations in bedload transport rates and storage in
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Hoey, …. 2000?
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Jagers, H.R.A. (2003) Modelling Planform Changes of Braided Rivers. PhD Thesis.
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University of Twente. The Netherlands.
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Murray, A.B., Paola, C. (1996) A new quantitative test of geomorphic models,
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Murray, A.B., Paola, C. (1994) A cellular model of braided rivers. Nature, 371, 54-
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Murray, A.B. (2003) Contrasting the goals, strategies, and predictions associated
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Paola, C. (2000) Modelling stream braiding over a range of scales. Gravel-bed rivers
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Parker, G. (1984) Lateral bed load on side slopes, Discussion. J. of Hydraulic
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Robertson-Rintoul, M.S.E. and Richards, K.S. (1993) Braided channel patterns and
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Sapozhnikov, V., Foufoula-Georgiou, E. (1996) Self-affinity in braided rivers.
8
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Zarn, B. (1997) Einfluss der Flussbettbreite auf Wechselwirkung zwischen Abfluss,
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12
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14
Doeschl et al.
46
1
2
Flume dimensions :
11.5 m × 2.9 m
Average bed slope:
0.012
Introduced discharge:
1.5 litres / second
Introduced sediment
varied from ?????
discharge:
Median grain size:
0.07 cm
Cell length in raster DEM:
3.75 cm
3
4
Table 1: Information from the physical flume experiment
5
6
Doeschl et al.
47
1
Measure
Total channel width
Empirical estimate Laboratory river Modelled river
280 mm
620 mm
710 mm
2.8
5.3
0.43
1.65
2.9
3.4
19.3
8 mm
3.18 mm
0.99+ mm
Avg. max. scour depth
-14 mm
-21.22 mm
-1.75 - mm
Topographic variation
4.9
4.8
4.4-
28 mm
32 mm
130 mm
3 hours
30 hours
Avg. number of channels
per cross-section
Avg. number of
confluences per meter
Total sinuosity
Avg. max. bar height
Total link length / number
of channel confluences
Time to develop to fully
braided state
2
3
Table 2: Characteristic scales estimated from empirical studies and observed in the
4
laboratory and modelled river, respectively. The superscripts + and – indicate and
5
increasing / decreasing tendency with time. For the empirical estimates of the above
6
measures the following formula were used: total channel width: Van den Berg
7
(1995); total sinuosity: Robertson-Rintoul and Richards (1993); average maximal
8
scour depth / bar height as well as topographic variation: Zarn (1997); total link length
9
/ number of confluences: Ashmore (2000).
10
11
Doeschl et al.
48
1
Figure captions:
2
3
Figure 1: Cumulative frequency distributions for natural and laboratory braided rivers.
4
Fig 1a: distribution of standardized link length. Fig. 1b: distribution of standardized
5
deviations of elevation from the median elevation. Fig.1c and 1d: cumulative
6
distributions of lateral slopes for areas above the elevation median (positive
7
elevations) and for areas below the elevation median (negative elevations) for the
8
Sunwapta River (c) and the laboratory river (d). Fig. 1a was published in Ashmore
9
(2000).
10
11
Figure 2: Deviations from the elevation median for cross-sections of the Sunwapta
12
River (a, b), the laboratory river (c, d) and the MP modelled river (e, f). The dotted
13
line represents the deviation of the elevation mean from the elevation median.
14
15
Figure 3: Digital elevation models of the topography and topographic changes of the
16
laboratory river. The overall slope was removed before plotting.
17
18
Figure 4: Digital elevation models of the discharge and topography of the MP
19
modelled river. The overall slope was removed before plotting.
20
21
Figure 5: Dependence of the average channel width (left) and the nodal spacing (right)
22
on the dimension of the computational grid in the MP modelled river.
23
24
Figure 6: Cumulative frequency distributions for natural, laboratory and MP modelled
25
braided rivers. Fig 1a: distribution of standardized link length. Fig. 1b: distribution of
Doeschl et al.
49
1
standardized deviations of elevation from the median elevation. Fig.1c and 1d:
2
cumulative distributions of lateral slopes for areas above the elevation median
3
(positive elevations) and for areas below the elevation median (negative elevations)
4
for the laboratory river (c) and the MP modelled river (d).
Doeschl et al.