General linear stability properties of monoclinal shallow waves

We analyze the linear stability of monoclinal traveling waves on a constant incline, which connect uniform flowing regions of differing depths. The classical shallow-water equations are employed, subject to a general resistive drag term. This approach incorporates many flow rheologies into a single setting and enables us to investigate the features that set different systems apart. We derive simple formulas for the onset of linear instability, the corresponding linear growth rates and related properties including the existence of monoclinal waves, development of shocks, and whether instability is initially triggered upstream or downstream of the wave front. Also included within our framework is the presence of shear in the flow velocity profile, which is often neglected in depth-averaged studies. We find that it can significantly modify the threshold for instability. Constant corrections to the governing equations to account for sheared profiles via a “momentum shape factor” act to stabilize traveling waves. More general correction terms are found to have a nontrivial and potentially important quantitative effect on the properties explored. Finally, we have investigated the spatial properties of the dominant (fastest growing) linear modes. We derive equations for their amplitude and frequency and find that both features can become severely amplified near the front of the traveling wave. For flood waves that propagate into a dry downstream region, this amplification is unbounded in the limit of high disturbance frequency. We show that the rate of divergence is a function of the spatial dependence of the wave depth profile at the front, which may be determined straightforwardly from the drag law.

Figure 1

Diagram of the system under consideration, showing a flow of depth $h(x,t)$ and velocity $u(x,t)$, propagating down a fixed incline at angle $\varphi$ to the horizontal. The profile depicts a typical monoclinal traveling wave solution connecting two uniform flowing layers.

Figure 2

Example traveling wave solutions satisfying Eq. (9) with $\beta =1$ and $\mathit{\text{F}}=0.5$, for Chézy drag (solid blue) and granular drag (dashed brown). The separate panels show connections from $h_0=1$ upstream to different downstream depths: (a) continuous monoclinal waves with $h_{\infty }=0.5$, (b) shock profiles with $h_{\infty }=0.1$, and (c) flood wave solutions with $h_{\infty }=0$.

Figure 3

Dependence of far-field characteristics ${\lambda }_1$ (solid) and ${\lambda }_2$ (dash-dot) on $\mathit{\text{F}}$, for Chézy drag monoclinal waves with $h_{\infty }=0.5$ and $\beta =1$. Upstream values are plotted with orange curves, downstream with purple curves. The inlaid diagrams show the directions of propagation of the characteristics in the three distinct solution regimes given by inequalities (16) to (18) (continuous waves, shock waves, no solution). These regimes are separated by vertical dotted lines. The wave profiles plotted in the leftmost and middle diagrams are $h_0\left(\xi \right)$ at $\mathit{\text{F}}=0.6$ and 1.8, respectively, within the interval $\xi \in [-50,50]$.

Figure 4

Existence of monoclinal traveling waves (MTWs) in the $(h_{\infty },\mathit{\text{F}})$ plane for (a) Chézy drag and (b) granular drag. The blue and orange lines denote the boundaries of the labeled regimes, given by the inequalities (16) and (18), respectively. Solid lines show the $\beta =1$ case and dashed lines show the $\beta =1.05$ case in panel (a) and $\beta =1.2$ in panel (b).

Figure 5

Spectrum diagram for the continuous monoclinal wave with $\mathit{\text{F}}=0.5$, $h_{\infty }=0.5$ and the granular drag law given in Eq. (11). Different regions of the spectrum are indicated in (a), labeled I–IV according to the signs of the far-field spatial decay rates $\text{Im}\left(k_{-}\right)$ and $\text{Im}\left(k_{+}\right)$. Details of this labeling are given in the main text. The critical lines $\text{Im}\left(k_{-}\right)=0$ and $\text{Im}\left(k_{+}\right)=0$ are plotted with yellow and blue dashes, respectively. Since the spectrum is symmetric about $\text{Im}\left(\sigma \right)=0$, since $exp(ik\xi +\sigma t)$ is invariant with respect to the transformation $\sigma \mapsto \overline{\sigma }$, $k\mapsto -\overline{k}$ (where the overbar denotes complex conjugation), we have omitted the upper half-plane. Regions I and II make up the essential spectrum, from which we show a selection of example modes, plotting $u_1\left(\xi \right)$ with solid curves in panels (b)–(f). Arrows indicate the direction of the perturbation velocities in the far field. In (b) we also plot the base solution $u_0\left(\xi \right)$ (dotted). The exact growth rates of the computed modes, as indicated by crosses in panel (a), are $\sigma =$ (b) 0, (c) $-0.002-0.01495i$, (d) $-0.0135-0.1i$, (e) $-0.00667881-0.0267i$, and (f) $-0.018-0.034i$. Where two modes exist (b, e, f), the plotted curves are first orthogonalized with respect to the inner product defined by $\langle f,g\rangle ={\int }_{-\infty }^{\infty }\overline{f\left(\xi \right)}g\left(\xi \right)e^{-{(\xi /100)}^2}d\xi$.

Figure 6

Spectra for discontinuous traveling waves with granular drag and $h_{\infty }=0.5$. Panel (a) shows the case $\mathit{\text{F}}=0.55$, close to the boundary of and within the regime of discontinuous states. The labeling and color scheme match the conventions of Fig. 5. An example mode is plotted in (b), which shows the $u_1\left(\xi \right)$ field of the mode at the location labeled with a cross in (a), $\sigma =-0.0109-0.2i$ (solid). The velocity of the base traveling solution, $u_0\left(\xi \right)$, is also shown (dotted). Panel (c) plots the spectrum of an unstable case with $\mathit{\text{F}}=0.8$. In (d) we plot the $u_1\left(\xi \right)$ field of the undamped unstable mode whose imaginary growth rate matches the example in (b), in this case located at the cross in (c), $\sigma =0.0159-0.2i$ (solid). The velocity of the base solution is also given (dotted). In both panels (b) and (d), arrows indicate the direction of the perturbation velocity far upstream.

Figure 7

Dependence of the critical $\mathit{\text{F}}={\mathit{\text{F}}}_c$ for instability of traveling waves on a constant ($h$- and $u$-independent) momentum shape factor $\beta$ for (a) Chézy drag and (b) granular (solid) and viscous (dashed) drag. In panel (a) the asymptote at $\beta =1.125$ is also plotted (dotted).

Figure 8

Spatial variation of the mode amplitude function $R$, determined numerically via Eq. (52), for granular traveling waves with $h_{\infty }=0.5$. (a) Amplitude $R\left(\xi \right)$ at $\mathit{\text{F}}=0.5$ (blue), plotted alongside $u_1\left(\xi \right)$ (gray) for a spatially undamped mode with $\sigma =-0.014884-i$. (This growth rate corresponds to $k_{-}\approx 0.74$. Higher wave number modes also agree well with $R$.) (b) Mode amplitudes for waves in the discontinuous traveling wave regime, for $\mathit{\text{F}}=0.85$ (red), 0.7 (purple), and 0.55 (blue), approaching the critical $\mathit{\text{F}}\approx 0.547$, that marks the change between continuous and discontinuous states when $h_{\infty }=0.5$ [see Fig. 4]. The $u_1$ field for a corresponding mode in the $\mathit{\text{F}}=0.55$ case, with $\sigma =-0.0113-i$ ($k_{-}\approx 0.85$) is also plotted (gray).

Figure 9

Far-field amplitudes of (the dominant branch of) marginal eigenmodes, for $\mathit{\text{F}}=2.5$ flood waves with (a) Chézy (blue), (b) granular (orange), and (c) viscous (pink) drag. For each drag law, we compute $u_1$ from Eq. (21) and evaluate its amplitude at $\xi =X=-40$, which (at the given $\mathit{\text{F}}$) is sufficiently far upslope that the underlying traveling wave approximates a uniform layer. Also plotted in each case is a section of the corresponding asymptotic amplitude $|R_{-}|$ (dashed gray), given by Eq. (67), verifying convergence as $k\to \infty$.