1、1,CHAPTER 1: FELIX M. EXNER AND THE ORIGINS OF MORPHODYNAMICS,Felix Maria Exner was an Austrian researcher who was active in the early part of the 20th Century. His main area of interest was meteorology. At some point he became interested in the formation of dunes in rivers (Exner, 1920, 1925; see a
2、lso Leliavsky, 1966). In the course of his research on the subject, he derived and employed one version of the various statements of conservation of bed sediment that are now referred to as “Exner equations.” In addition, he made an important early contribution to 1D nonlinear wave dynamics.,Dunes i
3、n the Mississippi River, New Orleans, USA Image from LUMCON web page: http:/weather.lumcon.edu/weatherdata/audubon/map.html,2,DUNE ASYMMETRY,Dunes are rhythmic bedforms that are often seen on the beds of rivers. The upstream (stoss) side of a dune has a gentle slope, whereas the downstream (lee) sid
4、e has a steep slope. This characteristic asymmetry allows the determination of flow direction.In the image on the right, the flow direction is from top to bottom. Dunes migrate in the same direction as the flow, i.e. downstream.,Magnified view of previous image,Flow,3,DUNES IN THE RHINE DELTA, THE N
5、ETHERLANDS,Image courtesy A. Wilbers and A. Blom,4,DUNES IN THE LABORATORY,Flow,Flow,Dunes in a channel at St. Anthony Falls Laboratory, University of Minnesota, USA,Dunes in a channel at Tsukuba University, Japan. Image courtesy H. Ikeda.,5,EXNERS QUESTION: WHY ARE DUNES ASYMMETRIC?,The parameters:
6、 x = streamwise distance L t = time T = bed elevation L qt = volume total sediment transportrate per unit stream width L2/T p = bed porosity 1 g = acceleration of gravity L/T2 H = flow depth L U = depth-averaged flow velocity L/T Cf = bed friction coefficient 1The notation in brackets denotes dimens
7、ions: M denotes mass, L denotes length and T denotes time. The bed friction coefficient is defined such that b = CfU2, where b denotes bed shear stress M/L/T2 and denotes water density M/L3.,6,THE EQUATIONS,St. Venant shallow water equations:,Exners equation of conservation of bed sediment:,Relation
8、 between sediment transport rate and flow hydraulics:,These are statements of conservation of water mass and momentum in a 1D river (constant width).,Exners seminal contribution: if more sediment enters a reach than leaves, the bed elevation in the reach increases.,The phenomenon of sediment transpo
9、rt was poorly known in Exners time. Exner guessed that a higher velocity caused a higher sediment transport rate.,7,THE RESULTS,The details of the analysis are not considered here. The essential result is a nonlinear equation for the evolution of bed elevation of the general formwhere c() denotes a
10、positive wave speed of the bed that is an increasing function of bed elevation. The result is that any symmetric bedform sharpens in time to the asymmetry characteristic of dunes.,Image from Leliavsky (1966),8,FELIX EXNER AND MORPHODYNAMICS,The field of morphodynamics consists of the class of proble
11、ms for which the flow over a bed interacts strongly with the shape of the bed, both of which evolve in time.That is, the flow and the bed talk to each other. The flow field over the bed determines a pattern of variation of sediment transport rate. This variation changes the bed by erosion or deposit
12、ion of sediment. The changed bed now induces a changed flow field.,Felix Exner was the first researcher to state a morphodynamic problem in quantitative terms. The term “morphodynamics” itself evolved many decades afterward. This notwithstanding, Exner deserves recognition as the founder of morphody
13、namics.,9,DERIVATION OF EXNERS EQUATION OF SEDIMENT CONSERVATON,Recall that qt denotes the volume sediment transport rate per unit width and p denotes bed porosity (fraction of bed volume that is pores rather than sediment). The mass sediment transport rate per unit width is then sqt, where s is the
14、 material density of sediment. Mass conservation within the control volume with a unit width requires that:,/t (sediment mass in bed) = mass sediment inflow rate mass sediment outflow rate,or,or,qt,qt,10,1D SEDIMENT TRANSPORT MORPHODYNAMICS,While Exners 1D model successfully explains a) the tendency
15、 of dunes to migrate downstream and b) their tendency to become asymmetric, it does not explain the origin of dunes. Such an explanation requires a 2D analysis, and as such is not considered further here. The interested reader can refer to e.g. Engelund (1970), Smith (1970) and Fredsoe (1974).,Ooi R
16、iver, Japan. Image courtesy H. Ikeda,Dunes are one of many examples of fascinating morphodynamic problems requiring 2D or 3D approaches. For example, the pattern of meandering and sediment sorting in the river on the right cannot be explained with a 1D approach.So why is this book limited to 1D morp
17、hodynamics?,There are many interesting 1D problems. A surprising number of 2D problems can be reduced and explained using a 1D formulation. 1D morphodynamics is the gateway to 2D and 3D morphodynamics.,11,RIVER STRAIGHTENING: A PROBLEM IN 1D MORPHODYNAMICS,Consider a meandering river.,In 1D approach
18、es, the relevant streamwise coordinate is measured as an arc length down the channel centerline. The long profile of the river is as illustrated below. The fluctuations are induced by the meanders themselves.,12,RIVER STRAIGHTENING (contd.),Now suppose the bends between points A and B are cut off du
19、e to river straightening.,The elevation drop between A and B is the same as before straightening (at least initially), but the distance between A and B is much shorter. As a result bed slope is elevated between A and B compared to the reaches upstream and downstream.,13,RIVER STRAIGHTENING (contd.),
20、Now lets exaggerate the change in bed slope for the sake of illustration.,The bed slope is higher, the flow swifter and the sediment transport rate higher over the steepened reach.,Exner equation of sediment conservation over a reach with length x (set porosity p = 0 for illustration).,Bed elevation
21、 of the reach increases in time if more sediment enters than leaves.,14,RIVER STRAIGHTENING (contd.),The reach UA must degrade (bed elevation must decrease in time), as there is more sediment output than input.The reach between B and D must aggrade (bed elevation must increase in time), as there is
22、more sediment input than output.,The sediment transport rate is higher over the steepened reach.,15,RIVER STRAIGHTENING (contd.),The upstream part of the reach must degrade and the downstream part must aggrade until a new equilibrium profile is established.The above 1D picture is a simplified versio
23、n of reality: too much aggradation downstream will cause the river to avulse (jump channel). Nevertheless, it captures the essentials of the problem.,Initial long profile after straightening.,Evolving long profile.,Final equilibrium long profile: qt/x = 0 thus /t = 0,16,AN EXAMPLE: THE EAST PRAIRIE
24、RIVER, ALBERTA, CANADA,River straightening was popular in the 1960s and 1970s. The goal was flood protection: the steeper slope of the straightened reach allowed higher flow velocities, and thus conveyance of the same discharge at a lower depth. This short-term benefit was often outweighed by the lo
25、ng-term pattern of aggradation and degradation induced by straightening. The East Prairie River provides such an example. See Parker and Andres (1976) for more details.,Degrading upstream reach,Straightened central reach,Aggrading downstream reach,East Prairie River, Alberta, Canada Images courtesy
26、D. Andres,17,THE STRAIGHTENED WEST AND EAST PRAIRIE RIVERS, ALBERTA, CANADA,Straightened central reach,West Prairie River,East Prairie River,meandering,straightened,Image from NASA https:/zulu.ssc.nasa.gov/mrsid/mrsid.pl,18,REFERENCES FOR CHAPTER 1,Engelund, F., 1970, Instability of erodible beds, J
27、. Fluid Mech., 42(2), 225-244. Exner, F. M., 1920, Zur Physik der Dunen, Sitzber. Akad. Wiss Wien, Part IIa, Bd. 129 (in German). Exner, F. M., 1925, Uber die Wechselwirkung zwischen Wasser und Geschiebe in Flussen, Sitzber. Akad. Wiss Wien, Part IIa, Bd. 134 (in German). Fredsoe, J., 1974, On the d
28、evelopment of dunes in erodible channels, J. Fluid Mech., 64(1), 1-16. Leliavsky, S., 1966, An Introduction to Fluvial Hydraulics, Dover, New York, 257 p. Parker, G. and Andres, D., 1976, Detrimental effects of river channelization, Proceedings, ASCE Rivers 76 Conference, 1248-1266. Smith, J. D., 1970, Stability of a sand wave subjected to shear flow of low Froude number, J. Geophys. Res., 75(30), 5928-5940.,
