Square patterns and quasipatterns in weakly damped Faraday waves.

Pattern formation in parametric surface waves is studied in the limit of weak viscous dissipation. A set of quasi-potential equations (QPEs) is introduced that admits a closed representation in terms of surface variables alone. A multiscale expansion of the QPEs reveals the importance of triad resonant interactions, and the saturating effect of the driving force leading to a gradient amplitude equation. Minimization of the associated Lyapunov function yields standing wave patterns of square symmetry for capillary waves, and hexagonal patterns and a sequence of quasi-patterns for mixed capillarygravity waves. Numerical integration of the QPEs reveals a quasi-pattern of eight-fold symmetry in the range of parameters predicted by the multiscale expansion. PACS numbers: 47.35.+i, 47.54.+r, 47.20.Ky Typeset using REVTEX 1 When a fluid layer with a free upper surface is subjected to vertical oscillation, Faraday waves are observed [1,2]. In a large enough container and for low viscosity fluids, standing wave patterns of square symmetry are observed near threshold [3]. Based on amplitude equations that we derive below, the experimental observation of square patterns in the capillary-dominated regime is explained. We also predict that hexagonal and a sequence of quasi-patterns can be stabilized for the case of a sinusoidal driving force as a result of triad resonant interactions for mixed capillary-gravity waves [4]. To our knowledge, this is the first theoretical derivation that starting from a realistic model of the fluid shows that a quasi-crystalline pattern is a stable steady state, and corroborates the conjecture of Newell and Pomeau [5] on the existence of the so-called “turbulent crystals”. Pattern-forming instabilities occur in a variety of extended nonlinear systems. The emergence of spatial patterns close to onset of the instability can often be described by amplitude equations [2,6]. However, for near-Hamiltonian (or weakly dissipative) systems, there is no general agreement on how dissipation should be incorporated into the amplitude equation formulation. Previous work on Faraday waves was based on amplitude equations for a purely Hamiltonian system, to which linear and nonlinear damping terms were added by introducing a dissipation function [7,8]. In this approach, linear dissipative effects in the original system contribute only to linear damping terms in the associated amplitude equations, while nonlinear damping terms result entirely from nonlinear dissipative effects. Such an approach has contributed to the general belief that for near-Hamiltonian systems, nonlinear saturation of the linear instability does not occur if only linear dissipative effects are considered, and weak nonlinear dissipative or other higher order effects are needed for nonlinear saturation [2,9]. In this paper, we show that in the case of weakly damped parametric surface waves linear dissipative effects do contribute to the nonlinear damping terms in the amplitude equation, and that they alone can saturate the parametric instability. In addition, the experimental observation of square patterns in capillary-dominated regime is naturally explained without having to invoke poorly understood nonlinear dissipative effects, or higher order terms in the amplitude equation. 2 The basic difference with previous studies [7,8] is that although the bulk flow does remain potential, it is modified by a rotational viscous boundary layer near the free surface that has to be explicitly incorporated into the analysis [10–12]. When the thickness of the viscous boundary layer is small compared to the typical wavelength of the pattern, the weak effects due to viscosity can be taken into account by introducing effective boundary conditions for the otherwise potential bulk flow. This is the basic idea of the quasi-potential approximation introduced below. We first expand the equations governing the motion of an incompressible viscous fluid and the appropriate boundary conditions at the free surface in the small thickness of the free surface boundary layer, δ. The resulting equations are further simplified by recasting them in a nonlocal form that involves the flow variables on the free surface only; thus eliminating the need to explicitly solve for the flow in the bulk [12]. Let z be the normal direction to the surface at rest and g(t) = −g0 − gz0 sin Ωt the driving force where g0 is the constant acceleration of gravity, and Ω and gz0 are the angular frequency and the amplitude of the driving force respectively. We choose 1/ω0 ≡ 2/Ω as the unit of time and 1/k0 as the unit of length, with k0 defined by ω 2 0 = g0k0 +Γk 3 0/ρ, the linear dispersion relation for surface waves, where Γ is the surface tension and ρ the density of the fluid. Then the dimensionless, nonlocal and quasi-potential equations read [12], ∂th(x, t) = γ∇ h + D̂Φ−∇ · (h∇Φ) + 1 2 ∇(hD̂Φ) −D̂(hD̂Φ) + D̂ [ hD̂(hD̂Φ) + 1 2 h∇Φ ] , (1) ∂tΦ(x, t) = γ∇ Φ− (