Morphodynamic modeling of erodible laminar channels

A two-dimensional model for the erosion generated by viscous free-surface flows, based on the shallow-water equations and the lubrication approximation, is presented. It has a family of self-similar solutions for straight erodible channels, with an aspect ratio that increases in time. It is also shown, through a simplified stability analysis, that a laminar river can generate various bar instabilities very similar to those observed in natural rivers. This theoretical similarity reflects the meandering and braiding tendencies of laminar rivers indicated by F. Métivier and P. Meunier [J. Hydrol. 27, 22 (2003)]. Finally, we propose a simple scenario for the transition between patterns observed in experimental erodible channels.

Figure 1

Sketch of a riverbed: $h$ is the elevation of the sand surface and $d$ is the water depth. The axes $x$ and $z$ are tilted with respect to horizontal. The Saint-Venant approximation is used for the velocity field $\mathbf{u}=(u,v)$.

Figure 2

Different transport laws compared with the experimental results obtained by [14]. The grains are transported by a viscous flow in a circular Hele-Shaw cell. $N_p$ is the particle flux and $V_s$ is the settling velocity of a particle. Dashed: Threshold law proposed by [14]; $N_p{d}_s∕V_s=0.85\theta (\theta -0.12)\mathcal{H}(\theta -0.12)$. Solid line: Power-law fit; $N_p{d}_s∕V_s=5.13{\theta }^{3.75}$. This law will be used as an illustrative case in the present study.

Figure 3

Widening of a straight laminar channel through erosion, modeled with Eq. (18). Parameter values are ${ϵ}_a=0.1$, $\alpha =0.8$, and $\beta =3.75$. Solid lines: Numerical solutions of Eq. (18) at different times (with avalanches). Dashed line: Self-similar solution (without avalanches, see Sec. 3B) at $t_{*}=10$, $t_{*}=100$, and $t_{*}=1000$. The presence of avalanches seems to influence only a small zone near the bank. The main part of the river section tends to the self-similar solution in any case.

Figure 4

Linear growth rate $\sigma$ of bed instability in a laminar river, versus the corresponding nondimensional wave number $k$. The fixed parameter values are $\beta =3.75$, $\gamma =1$, and $S=0.0875$. The Froude number and aspect ratio are varied according to a straight river widening (see Sec. 4B and points ${\Sigma }_i$ in Fig. 5). Above: $R_1=20.3$ and $F_1=3.94$; middle: $R_2=35.0$ and $F_2=3.21$; below: $R_3=55.0$ and $F_3=2.71$. For each set ${\Sigma }_i=(R_i,F_i)$, the solid curve corresponds to the mode $n=1$, whereas the dashed one corresponds to the mode $n=2$. The successive dominance of modes provides an interpretation for the transition from alternate bars to braids observed experimentally in [12].

Figure 5

Stability diagram for a laminar channel. The domains (separated by solid lines) are named after the most unstable mode between $n=1$ and $n=2$. The parameters values are $\beta =3.75$, $\gamma =1$ and $S=0.0875$. The dashed lines represent the evolution of a straight river when the water level is imposed $[F=F_0{(R∕R_0)}^{3∕4}]$ or when the outflow is imposed $[F=F_0{(R∕R_0)}^{-3∕8}]$. The three points ${\Sigma }_i$ correspond to the three cases presented in Fig. 4.