Phenomenological model of weakly damped Faraday waves and the associated mean flow

A phenomenological model of parametric surface waves (Faraday waves) is introduced in the limit of small viscous dissipation that accounts for the coupling between surface motion and slowly varying streaming and large-scale flows (mean flow). The primary bifurcation of the model is to a set of standing waves (stripes, given the functional form of the model nonlinearities chosen here). Our results for the secondary instabilities of the primary wave show that the mean flow leads to a weak destabilization of the base state against Eckhaus and transverse amplitude modulation instabilities, and introduces a longitudinal oscillatory instability which is absent without the coupling. We compare our results with recent one-dimensional amplitude equations for this system systematically derived from the governing hydrodynamic equations.

Figure 1

This figure illustrates the weak destabilization of the base solution against transverse amplitude modulations due to the coupling to mean flows. Shown are the critical values for instability ${\mu }_{\text{crit}}$ arising from either the long-wavelength component of the mean flow velocity alone (“ms”) or the short-wavelength component alone (“mo”) as a function of the coupling parameter ${\beta }_1$. The base solution is stable above the corresponding lines. The figure shows that the “ms” component of the flow does not appreciably modify the stability threshold, whereas the effect of the “mo” component is to weakly destabilize the base solution with increasing ${\beta }_1$. The wave number of the base solution is $q=1.04$, and we have used $\gamma =0.1,\alpha =0.5$, and $\Gamma =0.8$. The parameter ${\beta }_1$ spans the range between no mean flow and the approximate value that corresponds to the weak viscosity but shallow layer experiments of Ref. 5. The values of ${\beta }_2$ and $\lambda$ depend on ${\beta }_1$ according to Eq. (122).

Figure 2

Stability diagram for (a) the order parameter model of Eq. (113) without mean flow, and (b) with ${\beta }_1=0.5$. Other values of the parameters used are $\gamma =0.1,\alpha =0.5$, and $\Gamma =0.8$. We show the Eckhaus line (dotted), TAM line (dashed), and the zigzig line (dash-dotted). We also show the neutral stability curve of the primary instability (solid line denoted by $N$), and a saddle-node bifurcation (thick solid line marked $S$). Only the left branch of the Eckhaus line is shown emanating from $\left(q=1,\mu =0\right)$. This line is parabolic near the critical point, but quickly bends to the right as shown in the figure. Hence the region of stability of the base solution against an Eckhaus instability is the region below the dotted line. Comparison of (a) and (b) shows that the mean flow decreases the regions of stability against Eckhaus modulations and to a small extent against transverse amplitude modulations. In (a), the region of stability against all perturbations is shown by the gray area. This region is not indicated in (b) since these solutions are unstable with respect to the oscillatory instbility [38].

Figure 3

Real (top) and imaginary (bottom) parts of the largest eigenvalue for ${\beta }_1=0.2$, $\mu =0.1318$, and $q=1.07$. The inset shows the region near $k=0$.

Figure 4

Critical eigenvectors corresponding to the oscillatory instability at ${\beta }_1=0.2$, $q=1.0$, $\mu =0.1318$, and $k=0.04$. We show the temporal evolution over its own period (vertical axis) to illustrate the nature of the critical modes. The eigenvectors are represented by the $x$ component of ${\mathit{u}}^{ms}$ (a,c) and the amplitude of $\psi$ (b,d). The evolution is obtained from the two critical eigenvectors $v_i$ of the complex matrix $A(q,k,\dots )$ with eigenvalues $\pm i\omega$. Left and right traveling (a,b) waves are given by the evolution of $v_1{e}^{iegat}$ and $v_2{e}^{-i\omega t}$, respectively. A standing wave is obtained (c,d) by superposition of the evolution from both eigenvectors scaled to equal amplitude. The actual solution for $\psi$ is obtained from the superposition of the plotted eigenvectors with the base solution, see Eq. (124).

Figure 5

Stability boundaries of the oscillatory instability for three values of ${\beta }_1$: 0.05 (dotted line), 0.2 (dashed line), and 0.5 (solid line). The thick solid line indicates the saddle node. Periodic solutions are unstable to the left of the curves. Comparison with Fig. 2 shows that all periodic solutions are unstable and the pattern is expected to be time-dependent.