equilibrium condition implies that the cross-section A does not
depend on time. In the case of negligible lateral inputs to the river
reach, Eqs. (1) and (3) simply require that water and sediment discharges
Q and Qs keep constant along the longitudinal coordinate,
while Eq. (2) under steady flow conditions reduces to (see [16])
Q2
A
d
dx
a
A
þ g
dH
dx þ
Q2
C2A2 Rh
¼ 0: ð4Þ
Independently from the character (subcritical or supercritical) of
the flow conditions, this equation is solved by imposing as boundary
condition the water stage at the downstream section. This level
has been determined by considering the water-stage relationship
provided by the hydro-meteorological service of the Environmental
Protection Agency of the Emilia Romagna region [http://www.
arpa.emr.it/sim/?idrologia/annali_idrologici] for the section of
Pontelagoscuro and assuming a negligible contribution of the
Mincio and Secchia tributaries to the flow discharge.
The cross-section geometry used in the calculation is that resulting
from the sections recently surveyed by AIPO [4]. Each section has
been divided into an active central region, conveying the ordinary
flow discharge, and into two floodplains, flooded by larger discharges.
The model was then run under fixed bed conditions and calibrated
by choosing the values of the Gauckler–Strickler friction
coefficient, Ks, that minimized the mean square root error between
computed and observed stage-discharge relationship at the section
S42 near Borgoforte (see Fig. 2). The values of Ks selected for the
active channel and the lateral expansion areas have been set equal
to 25 and 15 m1/3/s, respectively. Note that C = KsRh
1/6/g1/2.
Finally, for a given upstream water discharge, the upcoming
sediment flux has been assumed to be in equilibrium with the local
hydrodynamics. The value of Qs has been estimated by considering
a relatively straight river reach comprised between Sections 57A
and 59B (near the town of Calto) and computing the sediment
transport rate by means of the Engelund and Hansen relationship
[12]. Fig. 5 shows the sediment transport capacity computed with
a grain size equal to 0.35 mm, as indicated by the granulometric
analyses carried out by ADBPO [2], and considering the bed topography
surveyed in 2005 (Fig. 3).
4. Results and discussions
We carried out several numerical runs by varying the upstream
water discharge in the range 500–6000 m3/s. The equilibrium bed
configuration for each discharge and the associated sediment flux
has been determined by solving Eq. (4). Specifically, we first calculated
the longitudinal distribution of the water surface elevation
and of the cross-sectionally averaged velocity for each discharge
Q. The knowledge of the flow characteristics at any cross section
then allowed us to evaluate the corresponding longitudinal distribution
of sediment transport capacity Qs. The average value of Qs
in a relatively straight portion of the river, included between Sections
57A and 59B, was then used as the sediment input to the
study reach.
Fig. 6 shows the equilibrium bed topographies computed for the
various water discharges, as compared to the bed topography surveyed
in 2005. Note that, differently from Fig. 3, showing the thalweg,
Fig. 6 reports the longitudinal distribution of the mean bed
elevation of the active channel in each cross-section. The particularly
high peaks in mean elevation observed at x 485 km and
512 km correspond to peculiar Sections 44C and 53B (Fig. 2),
characterized by the presence of two lateral river branches, separated
by a wide vegetated central bar located at a relatively high
elevation.
In general, it appears that if the considered discharge is too low
(e.g., 500 m3/s) with respect to formative conditions, the river bed