On the equilibrium profile of river beds

Abstract

[1] 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 1-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.

Where river and tide meet: The morphodynamic equilibrium of alluvial estuaries

“…Figure a shows the equilibrium bed profiles calculated for different values of L b with a prescribed value of r B , in the limiting case of purely fluvial conditions ( a 0 =0). The picture emerging from this figure confirms the recent findings of Bolla Pittaluga et al []: at equilibrium, alluvial channels react to narrowing or widening of the cross section by modifying the flow depth and the bed profile so as to keep the total sediment discharge constant. In particular, widening channels at equilibrium experience a progressive seaward decrease of the free surface as well as a bed aggradation associated with sediment deposition, leading to a reduction of the flow depth.…”

“…In nature, neither fluvial forcing nor tidal forcing is steady: they undergo fluctuations associated with random variations of the hydrologic cycle, as well as with variations of the astronomical component of the tide. As a result, equilibrium may actually be defined only in statistical terms, which is not surprising in the light of the analogue problem extensively investigated in the fluvial case [see Bolla Pittaluga et al , , and references therein]. Also, note that the present analysis adopts a one‐dimensional approach.…”

“…Unless otherwise stated, for a given value of Q u , the width B u and the sediment load Q su of the upstream river have been determined through the empirical equilibrium relations derived by Wilkerson and Parker [] for single‐thread sand‐bed rivers. Note that the uniform flow conditions embedded by these relations are characterized by an equilibrium bed slope similar to that predicted by employing the Chézy friction relationship and the Engelund and Hansen total load predictor [e.g., Bolla Pittaluga et al , ]. A number of simulations have been also carried out by considering nonequilibrium values of either Q su or B u at the upstream boundary.…”

Morphodynamics of a Width‐Variable Gravel Bed Stream: New Insights on Pool‐Riffle Formation From Physical Experiments

“…We build from Snow and Slingerland (), Repetto et al (), Bolla Pittaluga et al (), Duró et al (), Blom et al (), and Ferrer‐Boix et al (), and begin our analysis with four assumptions: (1) a channel reach of at least 10–20 in length that has a well‐defined average bed surface slope; (2) statistical steady state conditions, defined as (a) bed topography adjustment rates that tend to zero across the reach and (b) comparability between the rates of total sediment supply ( Q ss ) and sediment flux ( Q sf ) between the upstream and downstream boundaries, respectively; (3) characteristic grain sizes of the bed surface D i that are tending to a spatially uniform condition; and (4) channel banks that change position at rates much less than those of bed elevation and bed surface sediment texture.…”

“…PRE1 experimental data are supplemented in Figure with corresponding data calculated for experimental results reported by Nelson et al (), field‐based results reported by Thompson et al (), and numerical simulation results reported by de Almeida and Rodríguez () and Bolla Pittaluga et al (; see supporting information S3 for details of how Δ w ( x ) and S local were calculated for each study). Nelson et al () provide experiments guided by the physical characteristics of the middle reach of the Elwha River, WA, USA; Thompson et al () provide field data for North Saint Vrain Creek, Rocky Mountain National Park, Colorado; de Almeida and Rodríguez () provide numerical simulations of the Bear River, AK, USA; and Bolla Pittaluga et al () provide numerical simulations of the Magra River, Italy, used to test theory for the equilibrium profile of riverbeds. Plotted data for de Almeida and Rodríguez (), Bolla Pittaluga et al (), and Nelson et al () are reported as steady state or equilibrium profile conditions.…”