On the equilibrium profile of river beds

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH: EARTH SURFACE, VOL. 9, 37 33, doi:./3jf86, 4 On the equilibrium profile of river beds M. Bolla Pittaluga, R. Luchi, and G. Seminara Received March 3; revised 3 November 3; accepted 3 November 3; published 4 February 4. [] Despite the wide spectrum of perturbations of flow and sediment transport experienced by rivers as a result of hydrologic variations, the paradigm of morphodynamic equilibrium has long been present in the geomorphological literature where it is traditionally associated with the semiempirical notion of formative discharge, whereby the unsteady forcing is taken as morphologically equivalent to some effective steady forcing. Here we investigate the mechanisms responsible for maintaining a quasi-equilibrium bed profile of a river reach sufficiently short to have no significant tributary inputs. More importantly, we assume the channel banks to be fixed, hence, the case we have in mind is that of rivers protected by levees which cannot respond to hydrologic forcing by changing their width like natural rivers. Employing a -D model of river morphodynamics, we first determine the equilibrium profile of the river reach for given steady forcing conditions and discuss the capability of this approach for interpreting bed profiles observed in the field by applying it to the terminal reach of the Magra River, Italy. Field observations turn out to be reasonably well fitted by the equilibrium profile associated with a steady effective discharge, which however differs from the typical formative discharge (mean annual flood) for natural channels with erodible banks. Finally, we clarify how fluctuations of the hydrodynamic forcing associated with the recorded historical sequence of hydrologic events of variable intensities have acted to maintain the river equilibrium. Citation: Bolla Pittaluga, M., R. Luchi, and G. Seminara (4), On the equilibrium profile of river beds, J. Geophys. Res. Earth Surf., 9, 37 33, doi:./3jf86.. Introduction [] Field evidence suggests that on relatively small timescales (of the order of years or decades) and in the absence of major anthropogenic effects, the average bed profile of rivers may be in quasi equilibrium, i.e., the mean elevation of the active bed, a function of the longitudinal coordinate x, does not experience significant temporal variations. The above observation is often translated into the notion of a graded river as one which has become adjusted in form to transport both water and sediment discharges supplied by its drainage over a period of years [Mackin, 948]. Given the practical and conceptual importance of this notion, it is not surprising that the subject of rivers in quasi equilibrium has attracted a great and long standing interest of the scientific community. A comprehensive review of the past research in this area would be prohibitive. However, in order to set the present contribution in the context of the literature, it may be convenient to outline the various lines of research undertaken to investigate quasi-equilibrium rivers. Department of Civil, Chemical and Environmental Engineering, University of Genoa, Genoa, Italy. Corresponding author: M. B. Pittaluga, Department of Civil, Chemical and Environmental Engineering, University of Genoa, Via Montallegro, 645 Genoa, Italy. (michele.bollapittaluga@unige.it) 3. American Geophysical Union. All Rights Reserved /4/./3JF86 37 [3] A first line of research has been developed primarily in the geomorphological literature: here quasi equilibrium is traditionally associated with the notion of formative discharge of a river [Wolman and Miller, 96; Williams, 978; Andrews, 98; Leopold, 994]. Wolman and Miller [96] define the formative discharge of a river as a theoretical discharge such that, if maintained indefinitely, would produce the same channel geometry as the natural long-term hydrograph: essentially, it is assumed that the unsteady forcing on the river is morphologically equivalent to some effective steady forcing. Indeed, natural rivers display spatial variations and temporal fluctuations of flow properties and bed topography acting on a variety of scales. The smallest scales are associated with the formation of free and forced bed forms: free bed forms arise from instabilities of the bed interface and are typically migrating features; forced bed forms are generated by secondary flows driven by factors such as meandering, variations of channel width, bifurcations, and confluences. Larger-scale fluctuations are associated with variations of the driving forces, namely water discharge and sediment flux, forced by a variety of possible events: floods, natural seasonal oscillations, slow processes of degradation-aggradation induced by natural variations of sediment supply or abrupt variations of channel geometry (e.g., meander cutoffs). Hence, any notion of morphodynamic equilibrium can only be approximate (quasi equilibrium) and must refer to the cross-sectionally averaged bed elevation rather than to its local value.

2 [4] A variety of methods have been proposed for estimating the channel-forming discharge, including bankfull and effective discharge, and discharge at a specified recurrence interval. Each of them can be calculated using a procedure based on field indicators. The bankfull discharge is defined as the flow discharge which fills the channel to the top of the banks [Williams, 978]. The top of the bank is determined through several field indicators which, however, are not necessarily of general applicability nor they are free from subjectivity [Wolman and Leopold, 957; Nixon, 959; Schumm, 96; Harvey, 969; Bray, 97; Riley, 97; Pickup and Warner, 976]. A specific recurrence interval discharge has been associated with the bankfull stage, typically corresponding to an annual flood recurrence interval of approximately to.5 years with the.5 year recurrence flood considered as a reasonable value [Leopold, 994]. However, note that this recurrence interval applies only to temperate rivers, being longer in dryland rivers and shorter in tropical rivers. Moreover, it does not seem to be verified in many rivers [Williams, 978]. Variations in the frequency of bankfull conditions have been found both among different streams and along a single stream [Harvey, 969; Petit and Pauquet, 997; Castro and Jackson, ] implying that the specified recurrence discharge often generates poor estimates of the formative discharge. Finally, the effective discharge is defined as the increment of discharge that transports the largest proportion of the annual sediment load over a period of many years [Andrews, 98; Emmett and Wolman, ]. It incorporates the principle introduced by Wolman and Miller [96], which states that the channel-forming discharge, a function of both the magnitude of sediment-transporting events and their frequency of occurrence, may be identified as the value of the discharge which maximizes the product of the flow frequency and the sediment transport rate. A large amount of flow and sediment data is required to estimate this value, which makes it difficult to exploit this notion in many cases. We will return to this point below, where we show that this definition proves useful when implemented by means of numerical simulations of the morphological evolution of river reaches. [5] Despite uncertainties and limitations, the paradigm of morphodynamic equilibrium, defined in some average sense, has proven instructive. In particular, the use of a representative channel-forming discharge has helped in hydraulic geometry theories to define the main morphological characteristics of alluvial rivers [Leopold and Maddock, 953]. Uniform steady state formulations are also at the basis of the so-called rational regime theories, i.e., physically based theoretical derivations of relationships between equilibrium channel characteristics (channel width, channel slope) for given discharge and sediment supplies [e.g., Parker, 978, 979; Parker et al., 7; Ferguson, 986; Eaton et al., 4; Buffington, ]. [6] A second line of research has concentrated on the search for a possibly universal shape of the longitudinal river profile. To describe its typically upward concave shape a variety of mathematical functions, e.g., exponential functions, power law functions and logarithmic functions, have been proposed. Early suggestions were based on empiricalsemiempirical arguments which assumed the dominance of a single variable as controlling factor for the equilibrium configuration of the stream profile: exponential profiles were derived from the assumption that grain size variation is the dominant variable [Sternberg, 875; Shulits, 94; Yatsu, 955], while empirically calibrated power law profiles were proposed employing the flow discharge as dominant variable [Gilbert, 877;Leopold and Maddock, 953]. Heuristic analogies with other physical systems were also employed. Some of them assumed that the system is governed by a diffusion equation [Scheidegger, 97] from which exponential profiles were derived, power law and logarithmic profiles were predicted through random walk approaches [Leopold and Langbein, 96] while stream energy assumptions [Langbein and Leopold, 964] led to power law profiles. [7] A third, more recent line of research relies on the use of mechanistic models employing the basic laws of water and sediment conservation to allow for a deeper understanding of the role of various mechanisms involved in the establishment of quasi-equilibrium states [Snow and Slingerland, 987; Sinha and Parker, 996; Morris and Williams, 997; Paola and Seal, 995]. [8] In particular, Snow and Slingerland [987] have employed a steady state approach to analyze the dependence of the stream profile on imposed downstream variations of flow discharge, sediment supply, channel width, and sediment caliber. These variations were derived from hydraulic geometry relationships. Hence, the approach of Snow and Slingerland [987] still relies on the notion of formative discharge and on semiempirical regime relationships to impose external variations of the controlling variables. Exponential, logarithmic, and power function curves were all found to fit the calculated bed profiles quite closely. Sinha and Parker [996] followed an approach similar to that employed by Snow and Slingerland [987] to investigate analytically the role of wavelike bed progradation, sediment abrasion, basin subsidence, and inflows from tributaries in driving the concavity of the bed profile. Finally, the effect of selective sorting as well as sediment abrasion was studied by Morris and Williams [997]. Their theoretical model showed that streams with low solids concentrations and low lateral inflows, with bed load sediments undergoing either selective sorting or sediment abrasion have exponential profiles. However, note that, by using Sternberg s [875] exponential law for abrasion, these authors imposed an exponential forcing on the system. The effect of selective sorting leading to downstream fining has also been investigated by Paola and Seal [995], who showed that it is typically associated with a long profile of the river that is concave upward. [9] In the present contribution we wish to further understand the mechanics of morphodynamic equilibrium. The issue we want to tackle here is to understand the mechanisms responsible for maintaining a quasi-equilibrium bed profile at the scale of a river reach. Our aim is then less ambitious than that of previous mechanistic attempts [Snow and Slingerland, 987; Sinha and Parker, 996; Morris and Williams, 997], where most of the ingredients of the real problem (spatial variations of grain size and flow discharge, temporal variations of channel width) were included in the formulation. On the contrary, we consider a river reach sufficiently short not to receive any water and sediment inflows from tributaries, thereby justifying the assumption that the grain size distribution is kept fairly uniform 38

3 throughout the reach. Moreover, we consider a river reach with fixed banks, hence, the channel width is externally forced rather than being an independent variable. These assumptions severely restrict the validity of our approach, and we do not pretend to fully interpret how natural rivers work. Indeed, in nature, rivers are able to respond to hydrologic forcing in other ways than simply modifying their bed elevation: in particular, they may change their width or their grain size distribution. [] Moreover, herein we are interested in river equilibrium on timescales of the order of decades. On larger temporal scales (of the order of centuries), the average bed profile may undergo aggradation or degradation in response to a variety of both natural and anthropogenic slow forcing, e.g., variations of sediment supply, sea level rise, and subsidence. A more general definition of equilibrium can be applied to systems which undergo net erosion or deposition: a steady state river profile may indeed be attained in this case provided the divergence of sediment flux is exactly balanced by the effect of subsidence and sea level rise. Our quasi equilibrium may be seen in general as a very small portion of a slow aggradational or degradational wave, in a similar fashion as a tidal wave, if observed locally and for a sufficiently short time, may be interpreted as a quasi-steady phenomenon. The dynamic character of equilibrium when the system is viewed on the larger temporal scale is typically exemplified by deltas, net depositional systems which have been widely investigated in the last two decades [Paola et al., ], and in particular by the response of coastal plain rivers to slow variations of the relative sea level [Parker et al., 8a, 8b; Swenson and Muto, 7]. [] Notwithstanding the above limitations, the problem of morphodynamic equilibrium for river reaches with fixed banks is of great engineering importance. Indeed, our work arose from a question posed to us by the local authorities of a river basin, who were pressed to dredge the terminal reach of the river to promote navigation, a practice opposed by environmentalists. The approach presented below was developed precisely to help the authorities take a decision: the notion of morphodynamic equilibrium that was needed to fulfill this goal is precisely that adopted in the present work. A discussion on the implications of the present investigations for the above issue is presented in section 4. Also note that, in this respect, our aim is more ambitious than those of previous attempts, in that we consider a real river configuration rather than an ideal one and we attempt to predict mechanistically the observed equilibrium bed profile, clarifying how that equilibrium is maintained through the morphodynamic response of the river bed to the sequence of flood events that periodically tend to disrupt the established equilibrium. As discussed below, this analysis proves instructive and suggests that the typical assumption that the formative flow is the bankfull discharge is unlikely to apply to rivers with fixed banks. [] In order to check the degree of success that may be achieved through our theoretical approach, it is crucial that the model be validated on a test case. Fortunately, the Magra River (Italy) (Figure ) provides a perfect example, which has a great advantage for the modeler, namely the availability of a number of systematic surveys of river topography performed in the last few decades. In this paper we show that the observed bed profiles are largely predictable. BOLLA PITTALUGA ET AL.: EQUILIBRIUM PROFILE 39 [3] We pursue the above aims through a series of steps. First, we formulate the problem of river morphodynamics in -D (section ). Next, we introduce a mechanistic definition of morphodynamic equilibrium (section 3) and outline a mathematical approach that is capable of determining the equilibrium profile of a river reach for given steady forcing conditions. For the sake of completeness, we then briefly summarize well-known results concerning the simplest, yet fundamental case of uniform width channels (section 3.). We then move to the general, more realistic, case of nonuniform width channels. We first treat the ideal case of rectangular channels with variable width (section 3.3), which is amenable to analytical treatment. Next, we test our general approach on the terminal reach of the Magra River (section 3.4) and show that observations are fairly satisfactorily fitted by the predicted equilibrium profile associated with some steady formative discharge. Finally, we clarify how fluctuations of the hydrodynamic forcing associated with the recorded historical sequence of hydrographs of variable intensities act to maintain such an equilibrium (section 4).. Governing Equations of -D Morphodynamics [4] In order to provide a mechanistic definition of morphodynamic equilibrium, it is convenient to start with the -D formulation of the problem of river morphodynamics. [5] Let us then consider a straight erodible channel (Figure ) and denote by (x, t) the cross-sectional area of the stream at the longitudinal coordinate x and time t. Moreover, let R(x, t) be the hydraulic radius, Q(x, t) the fluid discharge, h(x, t) the free surface elevation, and C(x, t) the dimensionless Chézy coefficient. The one dimensional form of the governing equations for the fluid phase may then be written in the @x = q l () + Q C R = m l () Here q l and m l are respectively the rate at which mass and momentum (per unit density) are supplied to the river per unit channel length either by tributaries or by the floodplain. Moreover, ˇ(x, t) is the coefficient accounting for the deviation of local values of fluid momentum from its cross-sectional average. The channel being erodible, we may also write = [h(x, t), (x, t), x] (3) having denoted by (x, t) the mean elevation of the active bed. Hence, d ˇ (4) d dx =,x h; @x @x. (5) [6] In order to complete the formulation of the morphodynamic problem, we need the conservation equations for

4 BOLLA PITTALUGA ET AL.: EQUILIBRIUM PROFILE Figure. (a) The Magra river basin and the reach investigated in the present work (dashed ellipse). (b) The middle Magra and (c) the microtidal estuary. the solid phase. The -D form of the continuity equation for the solid phase [Exner, 95] reads as follows: + (bf qs ) (6) with p the sediment porosity, bf (x, t) the bottom width, qs (x, t) the average sediment discharge per unit width, and qsl the lateral sediment discharge per unit channel length supplied by tributaries or by the floodplain. Below, we assume that the quantity qs (x, t) can be given the following general form: qs = qs [ (x, t); RP ] not treat transport in suspension distinctly from bed load, which would require the solution of an additional advectiondiffusion equation for the solid concentration. [7] Also note that the problem of morphodynamics is coupled to the hydrodynamic problem through equations (6) (8): the hydrodynamics of equation (8) determines the sediment transport capacity of equation (7) that in turn affects bed morphology as predicted by the Exner equation (6) and then modifies the flow hydrodynamics in a recursive process. (7) Here and RP are the average Shields stress and the particle Reynolds number, respectively: = Q C (s )dg RP = p (s )gd 3 (8) with s the relative sediment density and d the sediment size. Note that, assuming that sediment transport may be described through a general predictor described from equation (7) (which includes total load formulas adequate for sandy streams like Engelund and Hansen [967] s formula), the only assumption we are making is that we do Figure. Sketch of an erodible channel and notations. See also the notation list for definitions of the symbols. 3

5 [8] Finally, the equations (), (), and (6) define a hyperbolic system of partial differential equations whose solution requires the knowledge of appropriate initial and boundary conditions. [9] The initial conditions assign the values of the unknown functions, say h(x, t), (x, t), and Q(x, t), at time zero and any value of the spatial coordinate x throughout the reach under consideration. Hence, h(x, t) t= = h (x), (x, t) t= = (x), Q(x, t) t= = Q (x) ( < x < L) (9) [] The boundary conditions assign the values of the unknown functions at any time at the upstream (x = ) and/or downstream (x = L) ends of the reach under consideration, depending on the sign of the characteristic celerities, i.e., the eigenvalues associated with the governing system of equations (), (), and (6). It is well known that two conditions must be imposed at the upstream end and one condition is required at the downstream end. The precise choices of boundary conditions made in our simulations will be specified in section Morphodynamic Equilibrium 3.. Formulation of the Problem of Morphodynamic Equilibrium [] Strictly speaking, an erodible stream is in morphodynamic equilibrium provided its boundary does not undergo any temporal changes: hence, its bed should neither aggrade nor degrade (@/@t ), and its banks should neither retreat nor advance. As discussed in section, we restrict our attention on the former constraint and treat river banks as fixed. [] Strict equilibrium naturally also requires steady hydrodynamic conditions (@/@t ). If we also assume that no lateral inputs are present in the reach (q l = m l = q sl =), then the governing equations of morphodynamics presented in section reduce to the following form: A second, less trivial, condition would be that the total sediment discharge does not vanish but is kept spatially constant throughout the reach investigated: this is a less trivial dynamic condition which can be readily analyzed under steady conditions and proves quite instructive. We systematically investigate the consequences of the latter constraint and show that a distinct equilibrium state is associated with any given set of forcing conditions, namely given values of flow and sediment discharge. [4] For the sake of providing a systematic and complete picture of the problem, we will start by summarizing results for the simplest well-known case of uniform width channels. 3.. Morphodynamic Equilibrium of Uniform Width Channels [5] If the geometry of the channel is perfectly uniform (i.e., channels with cross section which does not vary in the longitudinal direction), then the bottom width b f is constant and the constraint given by equation () leads to the stricter requirement that the laterally averaged value of the total sediment flux per unit width q s (x, t) should also be constant. For a spatially constant grain size distribution, it then follows that the average Shields stress * must be kept constant throughout the reach. This condition, along with flow continuity (equation ), implies that the cross-sectional area and consequently the flow depth must also be constant. Recalling the momentum equation () we are led to the well-known conclusion that the channel slope must be kept constant, i.e., the flow must be strictly uniform. In other words, at equilibrium, the stream adjusts its slope and its flow depth to the driving forces Q and Q s. [6] In particular, for a straight, wide rectangular channel with constant width, using the Chézy relation to express the constraint given by equation () and a general transport capacity relationship to express the constraint given by the equation (), we may write Q =(bk s p S )Y 5 3u, Q s = b p (s )gd 3 n ( * *c ) m (3) Q = constant () Q s = b f q s = constant () Q d ˇ + g dh dx dx + Q = () C R This system must be solved with the help of some closure algebraic relationship for the average sediment flux per unit width q s (x, t) of the type (7). Finally, the knowledge of the given values of flow and sediment discharge replaces two boundary conditions, so that only one boundary condition for the unknown functions h(x) or (x) is required and may be assigned either upstream or downstream, independently of the subcritical or supercritical character of the stream. [3] Equation () immediately suggests that there are two simple conditions which would ensure the requirement of bed equilibrium. A first (trivial) condition is that the sediment discharge vanishes everywhere throughout the reach investigated: in other words, the average bed shear stress should nowhere exceed the threshold value for sediment entrainment. In sand bed rivers like our study site, this static condition is indeed met, but only at very low stages. 3 where k s is Strickler s parameter, Y u is the uniform flow depth, n is an empirical constant and m is an empirical exponent larger than one. Simple algebraic manipulations with the help of (8) then give S = c s q 6 7 " # 7 m + *c, Y u = c Y q 6 7 n " 3 7 m + *c# n (4) where is the bed load flux per unit width q s scaled by p (s )gd3, q is the fluid discharge per unit width and the constants c s, c Y are c s = k 6 7 s [(s )d ] 7, c Y = k 6 7 s [(s )d ] 3 7 (5) This is a well-known result: an increased sediment supply for given fluid discharge tends to steepen the channel [Shen, 97; Soni et al., 98]; on the contrary, an increased fluid discharge for given sediment supply tends to flatten the channel. Moreover, since Strickler s coefficient is weakly dependent on the grain size, the coefficient c s and consequently the equilibrium slope, increase as the grain diameter

6 (a).5 (b) f ϕ Width ratio b/b Width ratio b/b Figure 3. The functions f and ' are plotted in terms of the width ratio b/b for different values of the ratio *c / *. We have adopted m =3/corresponding to a Meyer Peter-Müller type of transport formula. increases. Of course, similar equilibrium relationships can be established for any pair of variables chosen among Y u, S, q, and q s as functions of the remaining two. [7] Natural channels are hardly characterized by a uniform width: the width as well as the shape of their cross sections vary significantly in the longitudinal direction. Does an equilibrium bed profile still exist? Can we predict it for any given set of forcing conditions, namely given grain size distribution, channel geometry, and specific values of the flow and sediment discharge per unit width? 3.3. Morphodynamic Equilibrium of Nonuniform Width Channels: The Simple Case of Rectangular Channels [8] The nonuniform character of channels with fixed banks is felt by their morphodynamic equilibrium, as they respond to channel narrowing or channel widening by modifying the flow depth so as to keep the total sediment discharge constant. Other important morphologic adjustments, such as grain size response in coarse-grained rivers [e.g., Dietrich et al., 989; Parker, 99] and bed form response in sand bed rivers, are not considered here. [9] We first illustrate the above mechanism by analyzing a simple configuration consisting of a rectangular channel with variable width. In this case the bottom width b f coincides with the width of the free surface b and we may simply write Y = h, = by (6) Also note that, as a consequence of the equilibrium constraint, the only independent variable left is the longitudinal coordinate x. In other words, under equilibrium conditions, every function is dependent on the spatial coordinate only. It is also convenient to define an average flow and sediment discharge characterized by an average channel width b, such that, for a given flow and sediment discharge, we may define an equilibrium slope S and a uniform flow depth Y, corresponding to a reference Shields stress *. [3] The mechanism mentioned above is then immediately illustrated by imposing the constraint given by 3 equation (). Adopting the general transport formula (3) with m constant, this constraint becomes * *c * *c m = b b (7) Simple manipulations with the help of (8) lead to the following relationship: 3 Y b 7 ( m ) = f 3 7 (8) Y b having denoted by f the following function: f =+ *c * " # b m b (9) The dependence of this function with ratio and Shields stress is plotted in Figure 3a. Differentiating (8), one eventually finds dy dx = Y db b dx ' () having set: ' = m *c *c *c * + *c b m * b () This equation shows that ' is invariably positive (Figure 3b), hence the sign of dy/dx is opposite to that of db/dx, a result independent of whether the channel is steep or mildly inclined. Note that this finding is also independent of the particular relationship employed to evaluate the sediment flux. Hence, the equilibrium depth increases if the channel narrows and it decreases if the channel widens. The function ' will prove useful also when the effect of width variations on the bed and on the free surface profiles at equilibrium will be examined.

7 (a) τ *c /τ * =. τ *c /τ * =.5 τ *c /τ * =. (b) τ *c /τ * =. τ *c /τ * =.5 τ *c /τ * =. Depth ratio Y/Y.5.5 Froude ratio F/F.5.5 τ *c /τ * τ *c /τ * CHANNEL NARROWING CHANNEL WIDENING CHANNEL NARROWING CHANNEL WIDENING Width ratio b/b Width ratio b/b Figure 4. Equilibrium values of ratios of (a) flow depth Y/Y and (b) Froude numbers F/F as a function of the width ratio b/b for different values of the ratio *c / *. We have adopted m =3/corresponding to a Meyer Peter-Müller type of transport formula. [3] The relationship (8) has some further interesting implications. Denoting by F the Froude number, one finds that F 4 b 7 7m 9 9 = f 7 () b F To illustrate the significance of (), it is convenient at this stage to choose a transport relationship. We adopt a Meyer-Peter and Müller [948] type of transport formula and set m =3/. The equation for the Froude number () then becomes F b ( " 7 = + #) 9 7 *c b 3 (3) b * b F [3] As expected, the function F( b b ) is found to depend on the Shields stress (Figure 4). It is then of interest to consider two limiting cases: [33] Low Shields stress: c! [34] In this limit, (3) becomes F F 4 b 7 = (4) b From (4) it follows that a subcritical flow (F <) in a mobile bed channel at equilibrium remains subcritical (F <) at a channel narrowing, whereas a supercritical flow (F >) remains supercritical (F >) at a channel widening. [35] High Shields stress: c! [36] In this limit, (3) becomes so as to satisfy the constraint of constant sediment flux. It is also worth mentioning that these results apply to sufficiently gradual channel narrowing-widening such that flow separation does not occur and one dimensional modeling is appropriate. [38] Let us next examine the effects of channel narrowingwidening on the bed and free surface profiles at equilibrium. We then need to solve the momentum equation. Let us denote by (x) the component of free surface elevation driven by width variations and write h = h Sx + (x) (6) where h is the free surface elevation at the cross section x = where the perturbation takes the value. With the help of (6), the momentum equation () becomes an ordinary differential equation for the unknown function (x), which reads d dx = S Q + Q C gys g b db ( ') (7) dx having assumed that the cross section is sufficiently large to allow approximating the hydraulic radius by the flow depth and taking the Coriolis coefficient for fluid momentum ˇ(x) equal to one. Simple algebraic manipulations then allow us to reduce (7) to the following form: " d F dx = S F Y Y # /3 + Q db ( ') (8) g b dx F F b 7 = (5) b The trend predicted by (5) is opposite to that described above: a subcritical flow (F <) in a mobile bed channel at equilibrium remains subcritical (F <) at a channel widening, whereas a supercritical flow (F >) remains supercritical (F >) at a channel narrowing. [37] Note that these conclusions differ significantly from those that one would reach in the fixed bed case: indeed, an erodible stream at equilibrium chooses its own cross section 33 With the help of () and (8), and replacing the undisturbed slope S by the quantity F /C with C being the dimensionless Chézy coefficient, this equation is readily integrated in the following form: L = + F Y C Y I + I (9) Here the quantity I accounts for the perturbation of frictional terms induced by variations of flow speed and flow depth in the widening and narrowing reaches, while I

8 accounts for the convective contribution in the momentum equation (). I and I are dimensionless quantities and read Z Ox h i Z Ox I = Ob 6 7 7m f 7 dox, I = Ob dob 7m f 7 ( ')dox dox (3) BOLLA PITTALUGA ET AL.: EQUILIBRIUM PROFILE where Ox and Ob are the longitudinal coordinate and the channel width scaled by the length of the reach L and the undisturbed channel width b, respectively. [39] The solution (9) deserves some comments. Without loss of generality, we may take =, assuming that the cross section x = is located in the undisturbed constant width reach of the channel (either upstream or downstream). [4] Not surprisingly, the perturbation of the free surface is a fraction of the undisturbed flow depth which scales with the square of the Froude number F : hence, the effect of width variations on the free surface of subcritical streams is negligible. [4] The first term of equation 9 (involving I ) is driven by the perturbation of frictional terms induced by variations of flow speed and flow depth in the widening and narrowing reaches: its effect depends on the variations of channel width as well as on the intensity of sediment transport. This is readily illustrated. Assume that m = 3/ and consider two limits. [4] In the limit of weak sediment transport ( c! ), one readily shows that: [ ( b b ) 6 7 7m f 7 ]! ( b b ) 6 7. Hence, in the weak sediment transport limit, the contribution of the integral I is negative where the channel is wider than the undisturbed channel ( b b >) and vice versa. [43] The opposite trend is obtained in the limit of strong sediment transport ( c! ). In this case: [ ( b b ) 6 7 7m f 7 ]! ( b b ). Hence, in the limit of strong sediment transport, the contribution of the integral I is positive where the channel is wider than the undisturbed channel ( b b >) and vice versa. [44] The second term of the solution (9), involving I,is driven by perturbations of kinetic head induced by variations of flow speed and flow depth in widening and narrowing reaches. This term has the sign of db/dx: indeed, simple analysis of the definition () readily shows that ' <for any value of m (> )and ; similarly, from (9), it follows that the function f is invariably positive. Hence, at equilibrium, the effect of the second term is to let the free surface elevation rise when the channel widens and vice versa. [45] To make the above arguments quantitative, let us consider the rectangular channels depicted in Figure 5, representing examples of channels undergoing widening and narrowing, such that, in each case, the channel eventually recovers its original width b. More precisely, let us assume that Ob =+ ı [ cos( Ox)] with Ox [, ] and ı dimensionless amplitude of the width variation. [46] Figure 5 shows that, for the case considered, widening channels at equilibrium experience rising of the free surface and, vice versa, in narrowing channels, the free surface lowers. Is this invariably true? In order to answer this question, we have plotted in Figure 6 the quantities I Ox=/ and I Ox=/ as functions of the parameters and ı. [47] It appears that, as expected, the integral I Ox=/ has the sign of ı. On the contrary, the integral I Ox=/ may have the Elevation [m] (a) Width [m] Elevation [m] (b) Width [m] Flow -5 Free surface h Bed elevation η Flow Longitudinal coordinate x [m] -4 Free surface h Bed elevation η Longitudinal coordinate x [m] Figure 5. Sketch of two examples of channel undergoing (a) widening, (b =3m) and (b) narrowing, (b =5m) in reversed sequences. Input data: S = 3 ; Q = 5 m 3 /s; d =.m. same sign or the opposite sign depending on the intensity of sediment transport. However, Figure 6 suggests that I Ox=/ is at least an order of magnitude smaller than I Ox=/.Ifthe L factor is O(), then the modulus of the first term in (9) C is smaller Y than the second, i.e., widening channels at equilibrium experience rising of the free surface and vice versa. Note that this result suggests a well-known concept of flood conveyance: local widening of alluvial channels is not an appropriate solution to reduce the risk of flooding. [48] A second interesting feature of this solution is that, for symmetric width perturbations, the net effect of the second term vanishes (I Ox= = ), while the net effect of the first term does not vanish and I Ox= = I Ox=/. Hence, except for the case of intense sediment transport, in channels wider than the undisturbed reach, the first term is negative and leads to net lowering of the free surface downstream or net rising upstream; conversely, net rising downstream or net lowering upstream will result in channels undergoing spatial narrowing. In each case, whether the net effect is felt upstream or downstream depends on the boundary condition. [49] Finally, let us determine the bed elevation. Itis again convenient to introduce a perturbation e(x) of the uniform bed profile by defining 34 = Sx + e(x) (3)

9 (a) τ * = (b).4. τ * = I I NARROWING WIDENING - NARROWING WIDENING Figure 6. The integrals I Ox=/ and I Ox=/ as functions of the Shields parameter and the amplitude of width variations ı. where = h Y. With the latter definition and recalling the solution (9), one finds L e(x) =(x)+y Y(x) = + F Y C Y I + I h i + Y Ob m 3 f 3 7 (3) This solution suggests a simple result: in widening channels at morphodynamic equilibrium, the bed profile experiences aggradation and the bed elevation increases faster than the free surface elevation so as to allow for the predicted reduction of flow depth. Conversely, in narrowing channels at equilibrium, the bed undergoes degradation. [5] Next, let us consider a second instructive example, namely an estuary whose width increases in the downstream direction. The channel width b is assumed to vary exponentially in the landward direction according to the following relationship: x L b = b u +(b b u )exp (33) where L b is the channel convergence length, b is the width at the inlet and b u is the river width asymptotically reached upstream (Figure 7). Furthermore, we assume that the sea level h L, the flow discharge Q, and the average channel slope S are known. In Figure 7, we show the equilibrium configuration of the bed and the corresponding free surface elevation for a specific case. [5] These results show that, in order to accommodate the flow and sediment discharge prescribed from upstream, the equilibrium flow depth must decrease as the width increases, leading to a negative bed slope close to the estuary mouth. Such a tendency of the bed profile is typically observed in many microtidal estuaries, (e.g., in the estuary of the Magra river, Figure and Figure 8) and is exclusively associated with river widening close to the estuary mouth Morphodynamic Equilibrium of Natural Channels [5] We next attempt to ascertain whether our steady equilibrium paradigm does indeed help us in interpreting field observations. The tools needed to pursue this goal have L b 35 been presented in section 3.. We apply the latter approach to the terminal reach of the Magra River of length 6.4 km. Details of the general characteristics of the Magra catchment can be found in Surian and Rinaldi [3]. It suffices here to mention that the catchment, located in the Northern part of Tuscany, a region of central Italy, has an area of 7 km and consists of two distinct basins, the Vara River (length of about 65 km, area of 6 km ) on the west side and the upper middle Magra on the eastern side (length of about 54 km, Figure b). The two basins merge into the lower Magra (length of about 6 km) which forms a microtidal estuary ending into the Ligurian sea, characterized by very small tidal oscillations (amplitude around 5 cm, Figure c). The climate of the area is temperate, with a mean (maximum) annual precipitation around 7 (3) mm. The mean of the maximum annual daily discharge recorded at the (a) Elevation [m] m.s.l. (b) Width [m] Flow Free surface h Bed elevation η Longitudinal coordinate [m] Longitudinal coordinate [m] Figure 7. (a) The equilibrium configuration of the bed and the corresponding free surface elevation associated with an estuary whose width increases in the downstream direction with shape described by the relationship (33). (b) Plan form configuration. Data: b u = m; b = 3 m; L b = 5 m; L =, m; h L =m; S = 4 ; Q = 9 m 3 /s; d =mm.

10 Bed elevation [m] Sea Mean sea level Flow Longitudinal coordinate [km] Figure 8. Bed profiles of the terminal reach of the Magra River according to topographical surveys performed in 3, 8, and. Calamazza station (located on the Magra, upstream of the Vara-Magra confluence), draining an area of 93 km,is 683 m 3 s. The channel morphology varies from wandering in the upper basin (Figure b) to sinuous along the terminal reach (Figure c). [53] Below we will focus on the terminal reach of the Magra river (Figure c). Here detailed and systematic surveys have been performed. The reach displays significant width variations and has a sandy bottom characterized by a mixture of fine and coarse sand. No significant tributaries are present in the terminal reach, where the presence of fixed banks does not allow for lateral erosion during flood events. [54] The morphology of the Magra river has undergone significant changes in the first half of the twentieth century as a result of several anthropogenic factors, namely the construction of protection works, including a sequence of spur dykes in the middle reach, as well as few small dams and the exploitation of river sediments through extensive mining. As a result, the channel has narrowed and the braided pattern has slowly evolved into a transitional pattern, while strong bottom degradation occurred. The coastline underwent a significant regression, which stopped only around 95 when it had attained a configuration quite close to the present one [Surian and Rinaldi, 3]. [55] The bed profile of the terminal reach of the river has been quite stable between 3 and 8. Some changes of the bed profile are observed between 8 and as shown in Figure 8. Note that the longitudinal deformation of the bottom profile is mainly induced by the effect of width variations, with deposition favored by channel widening and scour driven by channel narrowing. It is important to note that, in the period 8, various flood events were experienced by the channel. Most notably, two large floods occurred in January 9 and December 9, characterized by estimated peak discharges of 355 m 3 s and 45 m 3 s, corresponding roughly to 5 and 3 year recurrence interval events respectively, in the lower Magra adjacent to the sea. An even larger flood was experienced in October, with an estimated peak of 4 m 3 s at the Calamazza station (located in the middle Magra not far from the Magra-Vara confluence) corresponding roughly to a year event and 5 m 3 s in the terminal reach, corresponding roughly to a 6 year event. In these reaches recovering equilibrium requires a longer period of low intensity events: it is then not surprising that some deviations from the 5 m 3 /s equilibrium profile are observed only within a short narrowest portion of the reach. They are the memory of previous flood events not yet forgotten. [56] We used our model to determine the steady formative discharge for the study site by choosing the flow and associated transport capacity that produce an equilibrium bed profile which best fits the observable stable river profile. Equation () was solved numerically for different values of the flow discharge Q, starting from the downstream end of the study reach where the free surface elevation was imposed equal to sea level. Simulations were performed using data for the cross sections surveyed in 3. In order to associate a transport capacity with a given flow discharge, we calculated the backwater profile for each discharge extending the calculation to a short reach upstream of the reach under investigation. The knowledge of the flow characteristics at any cross section in the upstream reach finally allowed us to evaluate the average transport capacity in the same reach. The average transport capacity of the short upstream reach was then used as the sediment input to the study reach. [57] We followed the approach proposed originally by Engelund [964], hence each cross section was divided into vertical panels assuming the cross-stream water level horizontal and the downstream slope constant across the entire section. Streamflow in each panel was then computed assuming locally uniform flow, and the total discharge in the cross section was calculated summing up contributions from all panels. Note that the approximation of locally uniform flow simply means that we are assuming that the velocity distribution is logarithmic. This approximation is commonly adopted in -D open channel hydrodynamics: nonuniformity affects the amplitude of velocity, while its effect on the shape of the velocity profile is ignored. [58] The sediment transport capacity associated with each flow discharge was evaluated using equation (3) with the observed value of the average bed slope in the terminal reach. The values of m and n were chosen as such to fit Engelund and Hansen s [967] formula and *c was set to zero. This choice was motivated by the well-known [Brownlie, 98] optimal performance of the Engelund and Hansen s [967] formula to predict the total load of sandy rivers where a significant portion of sediment is transported in suspension. The average value of the Strickler coefficient was assumed equal to 3 m /3 s and the average grain size was set equal to mm. Results of the calculation are reported in Figure 9. They suggest that the bed profile observed in 3 fits well the equilibrium profile associated with a discharge everywhere close to 5 m 3 /s, except for the reach where the river exhibits a strong constriction (located between longitudinal coordinate. km and 3.4 km). Here the 3 profile is closer to the equilibrium profile corresponding to a discharge of m 3 /s. As the assigned discharge increases, the equilibrium profile associated with that discharge deepens. [59] These results require a few comments. First, as stated above, although the bed profile has recently been quite stable (Figure 8), the available record of bed surveys in the reach investigated is not long enough to provide conclusive evidence that the 3 profile was indeed a quasiequilibrium profile. The fact that the 3 profile fits fairly closely an equilibrium profile predicted on a mechanistic basis suggests that this is likely to be the case. 36

11 - of sediment waves during flood events. This goal cannot be pursued in the context of a steady treatment of the problem. Bed elevation [m] Longitudinal coordinate [km] Figure 9. The equilibrium profiles of the terminal reach of the Magra River (Italy) predicted for various values of the flow discharge (black lines) are compared with the observed profile (red line), based on the detailed survey performed in 3. [6] Second, the value of 5 m 3 /s is close to the mean annual discharge for the study reach (347 m 3 /s), a value significantly lower than the mean annual flood discharge (7 m 3 /s), which is commonly used as an approximation of the bankfull and channel forming discharge [Leopold et al., 964; Knighton, 999]. As such, the above result conflicts with expectations based on prior studies of formative discharge. However, our calculated mean annual flood discharge may be erroneous due to the rather short time period (8 years) on which it is based, during which three very intense flood events have been experienced by the river, potentially causing the mean value to be overestimated. Moreover, a channel with fixed banks may have a formative discharge that differs from that of a natural channel with erodible banks. Indeed, in the latter case the channel width is the equilibrium width, while in the former case it is externally imposed: Why should the hydrodynamics and the associated morphodynamics be the same in the two cases? Our finding suggests that, on the contrary, the traditional notion of formative discharge defined as the mean annual flood may not apply to equilibrium conditions of rivers with fixed banks. [6] The third feature which requires some explanation is the deviation from the 5 m 3 /s equilibrium profile observed in the short narrowest portion of the reach (Figure 9). Is this feature a consequence of inaccuracy of the model? In the next section, we will show that deviations do not derive from inaccuracies of the model, but rather are the memory of previous flood events which had disrupted equilibrium and have not yet been forgotten by the bed profile. [6] In conclusion, the results of this section suggest that, despite the strong fluctuations of flow and sediment discharges undergone by the river, the bed profile has settled around an equilibrium state which is associated with steady forcing values of flow and sediment discharge falling in the lower range of values experienced by the river. It remains to be clarified how this equilibrium state survives the strong fluctuations of bed elevation driven by the propagation Morphodynamic Equilibrium and Unsteady Fluctuations [63] We have investigated how the bed profile of the river reach considered above has evolved in response to the sequence of flow events that occurred in the river throughout the period 3. This analysis has been performed solving numerically the full unsteady governing equations (), (), and (6) with the help of the explicit, second-order accurate in space and time MacCormack [969] scheme. In order to avoid spurious oscillations, arising from discontinuities of the solution in numerical schemes higher than first order [Godunov, 959], we have employed a total variation diminishing (TVD) algorithm that reformulates the MacCormack scheme [Garcia-Navarro and Alcrudo, 99]. The time step has been chosen to satisfy the classical Courant condition. [64] The numerical solution has been obtained for the terminal reach of the river of length 6.4 km. The computational grid was chosen such that nodes would coincide with the locations of the river cross sections surveyed in 3. [65] The initial condition employed in the simulation was the bed profile derived from the 3 survey, along with the free surface elevation obtained calculating the backwater curve associated with the initial value of the flow discharge. [66] The following boundary conditions have been adopted: [67] Initial Cross Section [68] The historical series of free surface elevations recorded between January 3 and January at Ponte della Colombiera were kindly provided by ARPAL (the regional Environmental Agency of Liguria): using a rating curve constructed through fixed bed calculations, these data were then transformed into a continuous hydrograph which was imposed as a boundary condition at the initial cross section. [69] A second boundary condition was obtained associating a sediment transport capacity with the above hydrograph and with the observed average slope of the bed profile in the upstream reach. [7] End Cross Section [7] The sea surface elevation was imposed at the end cross section, except during major flood events which are typically accompanied by coastal storms: during these events, a.5 m effective lifting of the free surface was assumed to account for the effect of wave setup. [7] With the latter conditions, simulations predicted the temporal evolution of the bed and free surface profiles in response to the imposed hydrograph. Results reported in Figure suggest a number of significant observations. [73] The simulation shows that the river reach, subject to the sequence of flow events that occurred in the period 3 8, undergoes an evolution of the bed profile which, starting from the 3 initial condition, reaches a state in agreement with the equilibrium configuration predicted for a formative discharge roughly equal to 5 m 3 /s (Figure a). Some discrepancies are found: in the very final reach close to the sea, where wave action interferes with the fluvial