WATER RESOURCES RESEARCH, VOL.27, NO. 1,PAGES 37-52, JANUARY 1991 Growth of Alternate Bars in Unsteady Flow MARCO TUBINO Hydraulics Institute, University of Genoa, Genoa, Italy A theoretical model isformulated toinvestigate the development ofthe amplitude of alternate bars in unsteady flows. The problem istackled by means of a weakly nonlinear analysis developed in a neighborhood ofthe threshold conditions for bar formation. Bar response tounsteady flow isfound to depend ona parameter •r thatis a measure of the ratio between the time scale of thebasic flow and the time scale ofbar growth. The present theory shows that if •r is0(1),as often occurs innature, flow unsteadiness affects the instantaneous growth rate and phase ofbar perturbations and controls the final amplitude reached by the bed configuration. A procedure fordetermining the final amplitude for a given flood event is proposed. Flume experiments were performed to test the main theoretical results. Thebed response to unsteady flow was measured for different values of the period of theflood. The observed temporal behavior of the bar amplitude proves to be strongly affected by the unsteady character of the flow for• of 0(I), as predicted bythe theory. INTRODUCTION The processof formation and development of migrating alternate bars has been the subjectof severalinvestigations starting from the cornerstone contributions by Leopoldand Wolman [1957] and Kinoshita [1961]. This problem bears both a conceptual and practical interest. It is well known that under appropriate conditions the flat cohesionless bottom of a turbulent streamflowingin a straight channel loses stability and bottom perturbations arise and develop spontaneously as a result of an instability process leadingto an alternating sequence of deep and shallowreaches.The formationof this topography is one of the basic processes controlling the channeldevelopment on a spatial scale of the order of the channel width and may play a relevant role in the process of meander formation [see Blondeaux and Seminara, 1985; Seminara and Tubino, 1989]. The practical motivation for investigating this process refers to the problem of formation of migrating bars in reaches of rivers straightened by regulation works such as channelization.Bar development needs be controlled since the related scouring and depositional effects can affect several aspects of fluvial engineeringlike navigation, bank protectionand designof structures. Many data have been collected on the occurrence of bars both in the field and in the laboratory and on the geometrical and hydraulic characteristics of alternate bars under steady- flow conditions, startingfrom the first observations of Ki- noshita [ 1961] until the recent and detailed investigations of Fujita and Muramoto [1985]. Empirical criteria for alternate- bar formationand predictors for the equilibrium length and heightof bars are also available in the literature (see among othersIkeda [1984] and Kuroki and Kishi [1985]). On the theoretical side a rational interpretation of the physical process hasbeen developed through a largenumber of linear studies[Hansen, 1967; Callander, 1969; Engelund and $kovgaard, 1973; Parker, 1976; Fredsoe, 1978]. Linear theories seekthe conditions for incipientbar formation,the lineargrowthrate of perturbations and their wavelengths and wavespeeds under steady-flow conditions. In particular Copyright 1991by the AmericanGeophysical Union. Papernumber 90WR0! 699. 0043-1397/91/90WR-01699505.00 they show that for given values of the Shields parameter 0 andof the roughness parameter ds (grainsize scaled by flow depth) an instability process occurs, providedthe width ratio /• (i.e., the ratio between the half-channel width and the flow depth) exceeds a criticalvalue/3 c, and leads to the develop- ment of migrating alternate bars [Tubino, 1986; Colombini et at., 1987]. Linear studies result in an exponential growth of the bar amplitude and are obviously unable to predict a "finite equilibrium amplitude" of the bed perturbations.The latter was determined by Colombini et al. [1986, 1987]by meansof a weakly nonlinear analysis developed in a neighborhoodof the critical conditions for bar formation. The development in time of the bar amplitude was found to be governed by a nonlinear ordinary differential equation (of Landau-Stuart type), the solution of which shows that nonlinear effects inhibit the exponentialgrowth predicted by linear theory and lead to an equilibrium value asymptotically reached. An independentattempt to tackle the latter problem was also made by Fukuoka and Yamasaka [1985]. The nonlinear analysisof Colombini et al. [1986, 1987] has been found to predict successfully the equilibrium value and the temporalbehavior of the bar heightduringthe process of development under steady-flow conditions [see $eminara and Tubino, 1989, Figures 6 and 7]. The experimental evidencesuggests that the wavelength of bottom perturba- tionsrapidly attainsa fairly stablevalue and then undergoes minor variationssuch that the formative process of migrating alternate bars is mainly characterized by the temporal in- creaseof the wave height until the bar geometry reaches an equilibrium state: this condition allows one to define an appropriate time scaleof the development process of bars. The question then arises of how the above process is affected by the unsteady character of the flow, which is always characteristic of rivers. Notice that the nonlinear development of river bedforms in the unsteady regime experienced during flood propagationhas received some attention only in the case of dunes [Allen, 1976; Fredsoe, 1979, 1981; Nakagawa and Tsujimoto, 1983; Tsujimoto and Nakagawa, 1984]. No attempt appears to have beenmadein the literature to analyzethe unsteady response of megaforms of the alternate-bar type to a variable flow regime. In gravel-bed rivers the actual formative conditions for 37