Amplitude equation and long-range interactions in underwater sand ripples in one dimension

We present an amplitude equation for sand ripples under oscillatory flow in a situation where the sand is moving in a narrow channel and the height profile is practically one dimensional. The equation has the form $h_t=-ϵ(h-\overline{h})+\mathbf{(}{\left(h_x\right)}^2-1\mathbf{)}{h}_{xx}-h_{xxxx}+\delta {\mathbf{\left(}{\left(h_x\right)}^2\mathbf{\right)}}_{xx}$ which, due to the first term, is neither completely local (it has long-range coupling through the average height $\overline{h}$) nor has local sand conservation. We argue that this is reasonable and show that the equation compares well with experimental observations in narrow channels. We focus in particular on the so-called doubling transition, a secondary instability caused by the sudden decrease in the amplitude of the water motion, leading to the appearance of a new ripple in each trough. This transition is well reproduced for sufficiently large $\delta$ (asymmetry between trough and crest). We finally present surprising experimental results showing that long-range coupling is indeed seen in the initial details of the doubling transition, where in fact two small ripples are initially formed, followed by global symmetry breaking removing one of them.

Figure 1

The experimental setup: A narrow channel with a layer of sand is moved periodically in a plexiglass tank.

Figure 2

Temporal development of flat bed instability. (a) For the amplitude equation (3), $ϵ=0.03$ and $\delta =1$ with periodic boundary conditions. (b) In the experiment [side view of the central region $\left(36\mathrm{cm}\right)$ of the channel]. The length of each black bar is $1\mathrm{cm}$—note that the vertical scale is twice that of the horizontal. $T$ denotes the number of oscillations.

Figure 3

Number of ripples in a domain of size $L=200$ for $\delta =0$, 1, and 5 as function of $ϵ$. Also shown (dashed line) is the corresponding “limiting number,” $N_{\lim }=L{k}_e\left(ϵ\right)∕2\pi$ and the nearest integer value, $\mathrm{Int}\left(N_{\lim }\right)$, larger than $N_{\lim }$ (full line with ◻). The numerics were done with finite difference methods using a variable-step Runge-Kutta algorithm (MATLAB) for time integrations. Periodic boundary conditions were used, and for these long simulations (around ${10}^6$ time steps) it was important to use the conserving form (2) for all terms except the one proportional to $ϵ$ to avoid drift of the mean height.

Figure 4

(a) “Doubling” transition by approximately halving the stroke length from initially $90\mathrm{mm}$ (top). After some time a new ripple appears in each trough leading to an intermediate state with twice as many ripples, which may then coarsen to fewer ripples in the final state. The initial doubling of the ripple number happens generically going to a state with more ripples. Note the slight drift typical of systems with a spontaneously broken translational symmetry. Each black bar is $1\mathrm{cm}$ long. (b) Doubling transition of (3) when changing $ϵ$ from 0.025 to 0.1 for $\delta =1$. (c) Transition when changing $ϵ$ from 0.006 to 0.2 for $\delta =0$. When $\delta =0$ the profile is up-down symmetric and doubling cannot occur. Instead, the amplitude initially decays almost to zero whereupon a modulation of the amplitude sets in. The decaying peaks or troughs subsequently split leading to an increased number of ripples in the final state. In all cases, we only show part of the system with a few ripples to make the bifurcation scenario clear.

Figure 5

Initial development of the doubling transition in Fig. 4a. Initially, two ripples appear in all troughs, but only one grows at the expense of the other. Either all the right ones or the left ones.