Design of Parametrically Forced Patterns and Quasipatterns

The Faraday wave experiment is a classic example of a system driven by parametric forcing, and it produces a wide range of complex patterns, including superlattice patterns and quasipatterns. Nonlinear three-wave interactions between driven and weakly damped modes play a key role in determining which patterns are favored. We use this idea to design single and multifrequency forcing functions that produce examples of superlattice patterns and quasipatterns in a new model PDE with parametric forcing. We make quantitative comparisons between the predicted patterns and the solutions of the PDE. Unexpectedly, the agreement is good only for parameter values very close to onset. The reason that the range of validity is limited is that the theory requires strong damping of all modes apart from the driven pattern-forming modes. This is in conflict with the requirement for weak damping if three-wave coupling is to influence pattern selection effectively. We distinguish the two different ways that three-wave interactions can be used to stabilize quasipatterns, and we present examples of 12-, 14-, and 20-fold approximate quasipatterns. We identify which computational domains provide the most accurate approximations to 12-fold quasipatterns and systematically investigate the Fourier spectra of the most accurate approximations

Quasipatterns in a parametrically forced horizontal fluid film

International audienceWe shake harmonically a thin horizontal viscous fluid layer (frequency forcing Ω, only one harmonic), to reproduce the Faraday experiment and using the system derived in [31] invariant under horizontal rotations. When the physical parameters are suitably chosen, there is a critical value of the amplitude of the forcing such that instability occurs with at the same time the mode oscillating at frequency Ω/2, and the mode with frequency Ω. Moreover, at criticality the corresponding wave lengths kc and k′c are such that if we define the family of 2q equally spaced (horizontal) wave vectors kj on the circle of radius kc , then kj + kl = k′n, with |kj| = |kl| = kc , |k′n| = k′c .It results under the above conditions that 0 is an eigenvalue of the linearized operator in a space of time-periodic functions (frequencyΩ/2) having a spatially quasiperiodic pattern if q ≥ 4. Restricting our study to solutions invariant under rotations of angle 2π/q, gives a kernel of dimension 4.In the spirit of Rucklidge and Silber (2009) [29] we derive formally amplitude equations for perturbations possessing this symmetry. Then we give simple necessary conditions on coefficients, for obtaining the bifurcation of (formally) stable time-periodic (frequency Ω/2) quasipatterns. In particular,we obtain a solution such that a time shift by half the period, is equivalent to a rotation of angle π/q of the pattern

Can weakly nonlinear theory explain Faraday wave patterns near onset?

The Faraday problem is an important pattern-forming system that provides some middle ground between systems where the initial instability involves just a single mode, and in which complexity then results from mode interactions or secondary bifurcations, and cases where a system is highly turbulent and many spatial and temporal modes are excited. It has been a rich source of novel patterns and of theoretical work aimed at understanding how and why such patterns occur. Yet it is particularly challenging to tie theory to experiment: the experiments are difficult to perform; the parameter regime of interest (large box, moderate viscosity) along with the technical difficulties of solving the free-boundary Navier–Stokes equations make numerical solution of the problem hard; and the fact that the instabilities result in an entire circle of unstable wavevectors presents considerable theoretical difficulties. In principle, weakly nonlinear theory should be able to predict which patterns are stable near pattern onset. In this paper we present the first quantitative comparison between weakly nonlinear theory of the full Navier–Stokes equations and (previously published) experimental results for the Faraday problem with multiple-frequency forcing. We confirm that three-wave interactions sit at the heart of why complex patterns are stabilised, but also highlight some discrepancies between theory and experiment. These suggest the need for further experimental and theoretical work to fully investigate the issues of pattern bistability and the role of bicritical/tricritical points in determining bifurcation structure

Spatiotemporal chaos and quasipatterns in coupled reaction–diffusion systems

In coupled reaction–diffusion systems, modes with two different length scales can interact to produce a wide variety of spatiotemporal patterns. Three-wave interactions between these modes can explain the occurrence of spatially complex steady patterns and time-varying states including spatiotemporal chaos. The interactions can take the form of two short waves with different orientations interacting with one long wave, or vice verse. We investigate the role of such three-wave interactions in a coupled Brusselator system. As well as finding simple steady patterns when the waves reinforce each other, we can also find spatially complex but steady patterns, including quasipatterns. When the waves compete with each other, time varying states such as spatiotemporal chaos are also possible. The signs of the quadratic coefficients in three-wave interaction equations distinguish between these two cases. By manipulating parameters of the chemical model, the formation of these various states can be encouraged, as we confirm through extensive numerical simulation. Our arguments allow us to predict when spatiotemporal chaos might be found: standard nonlinear methods fail in this case. The arguments are quite general and apply to a wide class of pattern-forming systems, including the Faraday wave experiment

Small divisor problems in Fluid Mechanics. In memory of Klaus Kirchgässner

International audienceSeveral small divisor problems occuring in Fluid Mechanics are presented. Two of them come from water waves: 3D periodic travelling gravity waves, and 2D standing gravity waves. The last example comes from quasipatterns observed for thin viscous horizontal fluid layers periodically vertically shaked (Faraday type experiment)

Which wavenumbers determine the thermodynamic stability of soft matter quasicrystals?

For soft matter to form quasicrystals an important ingredient is to have two characteristic lengthscales in the interparticle interactions. To be more precise, for stable quasicrystals, periodic modulations of the local density distribution with two particular wavenumbers should be favored, and the ratio of these wavenumbers should be close to certain special values. So, for simple models, the answer to the title question is that only these two ingredients are needed. However, for more realistic models, where in principle all wavenumbers can be involved, other wavenumbers are also important, specifically those of the second and higher reciprocal lattice vectors. We identify features in the particle pair interaction potentials which can suppress or encourage density modes with wavenumbers associated with one of the regular crystalline orderings that compete with quasicrystals, enabling either the enhancement or suppression of quasicrystals in a generic class of systems

Faraday instability on a sphere: numerical simulation

We consider a spherical variant of the Faraday problem, in which a spherical drop is subjected to a time-periodic body force, as well as surface tension. We use a full three-dimensional parallel front-tracking code to calculate the interface motion of the parametrically forced oscillating viscous drop, as well as the velocity field inside and outside the drop. Forcing frequencies are chosen so as to excite spherical harmonic wavenumbers ranging from 1 to 6. We excite gravity waves for wavenumbers 1 and 2 and observe translational and oblate-prolate oscillation, respectively. For wavenumbers 3 to 6, we excite capillary waves and observe patterns analogous to the Platonic solids. For low viscosity, both subharmonic and harmonic responses are accessible. The patterns arising in each case are interpreted in the context of the theory of pattern formation with spherical symmetry

Existence of Bifurcating Quasipatterns in Steady Bénard–Rayleigh Convection

Extending the results obtained in the sixties for bifurcating periodic patterns, the existence of bifurcating quasipatterns in the steady Benard-Rayleigh convection problem is proved. These are two-dimensional patterns, quasiperiodic in any horizontal direction, invariant under horizontal rotations of angle /q. There is a small divisor problem for q4.Using the results of Berti-Bolle-Procesi in 2010, we adapt it to a Navier-Stokes system ruling the Benard-Rayleigh convection problem. Our solution is approximated by the truncated power series which was formally obtained by Iooss in 2009, but which is divergent in general (Gevrey series). First, we formulate the problem in introducing a suitable parameter, able to move the spectrum of the linearized operator, as a whole, as for the Swift-Hohenberg PDE model. For using the Nash-Moser process, we are faced with the problem of inverting a linear operator which is the differential at a non zero point.There are two new difficulties: (i) First, the extra dimension leading to a more complicated spectrum of the linear operator. This first difficulty leads to use specific projections for reducing the spectrum of the studied operator, which we want to invert, to a finite set very close to 0. (ii) The second difficulty is the fact that the linearization L-(N) at a non-zero point leads to a non-selfadjoint operator, contrary to what occurs in previous works. This is more serious, and leads to use the spectrum of (LL(N)*)-L-(N) which depends mainly quadratically on the main parameter. A careful study of the bad setof parameters, with an assumption on the convexity of the eigenvalues of this operator, allows us to obtain a good estimate, as it is necessary for using the results of Berti etal. for solving the range equation. We again use separation properties of the Fourier spectrum (see the Bourgain and Craig results) for obtaining an estimate in high Sobolev norms. It then remains to solve the one-dimensional bifurcation equation.For any q4 , and provided that a weak transversality conjecture is realized, we prove the existence of a bifurcating convective quasipattern of order 2q, above the critical Rayleigh number