River bedform inception by flow unsteadiness: A modal and nonmodal analysis

River bedforms arise as a result of morphological instabilities of the stream-sediment interface. Dunes and antidunes constitute the most typical patterns, and their occurrence and dynamics are relevant for a number of engineering and environmental applications. Although flow variability is a typical feature of all rivers, the bedform-triggering morphological instabilities have generally been studied under the assumption of a constant flow rate. In order to partially address this shortcoming, we here discuss the influence of (periodic) flow unsteadiness on bedform inception. To this end, our recent one-dimensional validated model coupling Dressler's equations with a refined mechanistic sediment transport formulation is adopted, and both the asymptotic and transient dynamics are investigated by modal and nonmodal analyses.

Figure 1

Examples of morphologies relevant for the evolution of the river bottom: (a) river dunes after a flood event and (b) active antidunes [14]. The arrows indicate the stream direction. Bedform wavelength and amplitude are about $0.5–1$ m and $10–20$ cm, respectively.

Figure 2

Three possible linear evolutions of an initial, infinitesimal perturbation are reported in (a) monotonic asymptotic decay (A), transient growth before asymptotic decay (B), and monotonic indefinite growth (C). The dashed line indicates a schematic possible threshold of the perturbation amplitude beyond which nonlinear terms start to be non-negligible. (b) illustrates the concept of transient amplification on a two-dimensional system, where the nonorthogonality of the two eigenvectors ${\mathbf{e}}_{1,2}$ gives rise to transient growth: even though both eigenvectors, ${\mathbf{e}}_1$ and ${\mathbf{e}}_2$, individually decay in time, their nonorthogonal superposition causes the norm (length) of the resulting vector $\mathbf{e}$ to exceed its initial value before ultimately decaying to zero. Figures are qualitative and adapted from [21].

Figure 3

Scheme of a channel with river bedforms. The dotted and continuous lines refer to unperturbed and perturbed conditions, respectively. Note that the free-surface elevation is given by $H^{*}-D^{*}(1-\gamma )$.

Figure 4

The marginal stability curve (${\nu }_j=1$) is displayed for the steady (thick line) and the unsteady cases in the $F_0-k$ plane. The unsteady curves are obtained for $\omega =1$ and different values of $\varphi$.

Figure 5

Contour plot of optimal energy growth evaluated by the SVD for $\delta =0$ (steady case) in the $F_0-k$ plane. The line spacing is $\mathrm{\Delta }\widehat{G}=50$. The asterisk symbol marks the conditions chosen for the subsequent analysis.

Figure 6

Dune case ($\text{Fr}=0.7$): (a) behavior of $\widehat{G}$ as a function of the wave number $k$ for fixed times. The dashed line indicates the most unstable wave number for each time. The line spacing is $\mathrm{\Delta }t=100$. (b) Evolution of the most unstable wave number over time in conjunction with the corresponding value of the growth function.

Figure 7

Antidune case ($\text{Fr}=1.2$): (a) behavior of $\widehat{G}$ as a function of the wave number $k$. The dashed line indicates the most unstable wave number for each time. The line spacing is $\mathrm{\Delta }t=100$. (b) Evolution of the most unstable wave number over time in conjunction with the corresponding value of the growth function.

Figure 8

Growth function versus time, evaluated at the point $\{k,F_0\}=\{0.4,0.7\}$ for the steady system and for different values of the frequency $\omega (\varphi =0)$.

Figure 9

Growth function versus time evaluated at the point $\{k,F_0\}=\{0.4,0.7\}$ for the steady system and for different values of the phase $\varphi (\omega =0.1653)$.