On envelope dynamics in 2D Faraday waves

– A weakly nonlinear model for two-dimensional Faraday waves over infinite depth is derived and studied. Sideband instability of monochromatic standing waves as well as non-monochromatic solutions are studied analytically. Persistent irregular regimes are found numerically. Introduction. – Driven gravity-capillary waves is a well-known pattern-forming system owing its popularity to a relative simplicity and diversity of observed phenomena. Despite the wide interest in the problem, the theoretical basis of pattern formation in large-aspect-ratio systems is not yet fully understood. According to experimental data [1, 2] the complex dynamics are observed even at small supercriticalities. In the derivation of a theoretical model one must, therefore, consistently incorporate the effect of viscous dissipation, which governs the stability threshold and supercritical regimes. Furthermore, description of spatially non-homogeneous pattern formation involves “slow” space variables, whose scalings in a weakly supercritical domain depend strongly on the way how the dissipation was taken into account. Some of earlier models [2, 3] describing the envelope dynamics of driven waves in a largeaspect-ratio system have been derived from the equations for an inviscid fluid with damping incorporated phenomenologically by adding linear damping terms to the amplitude equations. As a result, an amplitude saturation due to nonlinear damping, which remains important even for low viscosity [4], is thus overlooked. In addition, the model obtained in [3] does not possess the symmetries of the underlying physical system. The nonlinear damping terms are included in some other models [5, 6]. However, due to neglecting of the boundary layer in the derivation of [5], those terms are of higher asymptotic order than in our derivation (see below). In spite of using the consistent multiple-scales method [6] with the viscous boundary layer rigorously accounted for, due to the chosen scalings the validity of the model seems to be limited to exceedingly small supercriticalities. In this letter we derive using results of [4] a single amplitude equation describing the evolution of one-dimensional patterns for small supercriticalities. This evolution equation is further investigated analytically and numerically. Model equations. – In a low-viscosity deep-water large-aspect-ratio system a packet of modes with adjacent wavenumbers becomes excited due to the parametric subharmonic resonance and locked to the (weak) forcing. In this case the principal part of a 2D motion of the free surface can be represented as a superposition of two counter-propagating small and narrow wave packets ζ(x, t) = âe + b̂e + c.c. (1) Typeset using EURO-LaTEX 2 EUROPHYSICS LETTERS Here ω is half the driving frequency and the corresponding wavenumber k̃ is obtained from the dispersion relation ω = gk̃ + (Σ/ρ)k̃, where Σ is surface tension, g is the gravity acceleration and ρ is the fluid density. In the case of weak forcing and low viscosity the slow spatio-temporal evolution of the envelopes â(x, t), b̂(x, t) is described by the set of equations [4] ∂tâ− s∂xâ = −(γ − 1 2 γ 3 2 )â+ if b̂+ (ic1 − p1 √ γ)|â|â+ (ic2 − p2 √ γ)|̂b|â, (2) ∂tb̂+ s∂xb̂ = −(γ − 1 2 γ 3 2 )b̂− ifâ− (ic1 + p∗1 √ γ)|̂b|b̂− (ic2 + p∗2 √ γ)|â|b̂, (3) where an asterisk denotes complex conjugate. The coefficients in eqs.(2),(3) are: a finite group speed s = dω(k̃)/dk, a small linear damping γ and the forcing amplitude f , and real-valued nonlinear dispersion coefficients, c1, c2, given by s = σ+ 1 2 , γ = 2νk̃ ω , f = f̂ 4 (1−σ), σ = Σk̃ 3 ρω2 , c1 = 3 4 σ− 3σ − 2 3σ − 1 , c2 = 3 2 σ+ 6σ + 4 3σ + 1 . Here f̂ is the amplitude of forcing (in units of g), σ describes the nature of the wave: it is a gravity wave or a capillary wave in two limiting cases of σ = 0 and σ = 1, respectively. Eqs.(2),(3) were derived in [4] from the Navier-Stokes equations using the multiple-scales method. Scalings for amplitudes and their slow variation in space and time were chosen as â, b̂ ∼ γ 1 2 , ∂xâ ∼ ∂tâ ∼ γ. Also, following [6] a stretched coordinate was introduced to resolve the boundary layer’s dynamics. The time and space variables have been non-dimensionalized using ω and k̃, respectively, as their scales. Higher-order corrections to the coefficients of the linear damping terms, γ 3 2 , and of the cubic terms, pi √ γ, represent, respectively, dissipation in the viscous boundary layer [8] and nonlinear interaction between potential wavefield and rotational flow in the boundary layer. Due to their cumbersome form, complex-valued coefficients pi are not presented here explicitly. Imaginary parts of the coefficients of the cubic terms are responsible for saturation of the amplitude growth by tuning the wave out of the resonance (nonlinear frequency shift). Asymptotically small coefficients of the cubic terms are retained in eqs.(2),(3) since nonlinear damping can become important for small supercriticalities [4, 9], for which a simpler model is derived below. A model similar to eqs.(2),(3) but without the higher-order damping terms was first derived using symmetry arguments and used in [2] with the coefficients estimated from the experiments. High-order nonlinear terms with real coefficients were included in [5]. However, as a result of neglecting of dissipation in the viscous boundary layer, the corresponding coefficients are asymptotically smaller (∼ γ) than in our model. In this paper we consider the case of p ≡ R(p1+p2) > 0, in which, as we show below, subcritical solutions do not exist. We also put aside the case of second-harmonic resonance, σ ≈ 1 3 , in which the coefficients c1, p1 diverge indicating a breakdown of the mode. Note, that the presence of the first spatial derivatives in eqs.(2), (3) follows only from the scaling ∂x ∼ ∂t ∼ γ ∼ f which is appropriate for the case under consideration. Indeed, according to the linear stability analysis [1, 4] the neutral curve for f ∼ γ ≪ 1 is defined by the equation s(1−k)+f c = f. Above the threshold fc = γ− 12γ 3 2 the bandwidth of unstable modes grows proportionally to f for a wide range of supercriticalities δ = (f − fc)/fc. Hence, the evolution of the envelopes takes place on the spatial scale f, which makes the spatial derivatives in eqs.(2),(3) of the same asymptotic order as the other terms. A similar “non-traditionally” slow spatial variation of amplitude takes place in a parametrically driven oscillatory dissipative system [7]. We relate the damping parameter and supercriticality as γ = γ 1δ , γ1 = O(1). The expansions ∂x = δ ∂x2 + δ ∂x3 + . . . , ∂t = δ ∂t2 + δ ∂t3 + . . . , {â, b̂} = δ 3 2 {a, b}+ δ 5 2 {a, b}+ . . . , are then introduced into eqs.(2),(3) and a hierarchy of problems is obtained and solved I.KELLER et al.: ENVELOPE DYNAMICS IN 2D FARADAY WAVE 3 at each order. At O(δ ) we obtain ∂t2a (1) = ∂x2a (1) = ∂t2b (1) = ∂x2b (1) = 0, a = ib. (4) At this order we find that the motion has a form of standing waves and the dynamics of their envelope evolve on the “slower” timeand space-scales t3, x3 . . .. At O(δ 9 2 ) we obtain using eq.(4) two equations ∂t3a (1) − s∂x3a(1) = −a + ib + ic|a|a, (5) ∂t3b (1) + s∂x3b (1) = −b − ia − ic|a|b, (6) where c ≡ c1 + c2. Adding eq.(5) and eq.(6) multiplied by i and using eq.(4) we obtain ∂t3a (1) = ∂t3b (1) = 0. A solvability condition of eqs.(2),(3) at order δ 11 2 yields an amplitude equation, which in terms of the unrescaled amplitude â and space and time variables x, t reads ∂tâ = (f − fc)â− p √ γ|â|â+ s 2 2f ∂ xâ+ isc1 f ∂x(|â|â) + isc2 f |â|∂xâ− c 2