Two-Mode Rhomboidal States in Driven Surface Waves

Arbell, H.; Fineberg, Jay

doi:10.1103/physrevlett.84.654

Cited by 60 publications

(90 citation statements)

References 26 publications

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“…• , we choose m = 4, n = 5 forcing: 4 : 5 forcing has been used in several experiments to produce 12-fold quasipatterns [1,19,33]. We setΩ(k = 1) = 2 so that the subharmonic response to frequency 4 comes at wavenumber 1, and we require that a wavenumber involved in 150…”

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confidence: 99%

Design of Parametrically Forced Patterns and Quasipatterns

Rucklidge¹,

Silber²

2009

SIAM J. Appl. Dyn. Syst.

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Abstract. 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 favoured. 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 stabilise quasipatterns, and 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.

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confidence: 99%

Design of Parametrically Forced Patterns and Quasipatterns

Rucklidge¹,

Silber²

2009

SIAM J. Appl. Dyn. Syst.

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“…To confirm our predictions for the pattern selection, we perform numerical simulations of the complex Ginzburg-Landau equation (6). Being interested in the formation of complex patterns comprised of (a) (b) Figure 13: Rescaled energiesF N for evenly spaced modes in the case K = 2 cos(tan −1 (2/5)).…”

Section: Numerical Simulations In Large Domainsmentioning

confidence: 99%

“…To determine the stability of the desired subharmonic patterns we derive the corresponding amplitude equations by performing a weakly nonlinear analysis of (6). We then use energy arguments to guide us in terms of the relative stability of the various pattern comprised of different numbers of modes.…”

Section: Weakly Nonlinear Analysismentioning

confidence: 99%

“…In particular in the Faraday system, in which a thin layer of fluid is vertically vibrated leading to patterns on the surface of the fluid, various kinds of superlattice patterns and quasi-patterns have been found experimentally depending on the frequency content of the forcing function [1][2][3][4][5][6][7][8][9]. Related complex patterns have also been observed in optical systems [10], in vertically vibrated fluid convection, where they arise from the competition of different instability mechanisms [11], and on the surface of ferrofluids driven by time-periodic magnetic fields, where they are due to spatial period-doubling [12].…”

Section: Introductionmentioning

confidence: 99%

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Superlattice Patterns in the Complex Ginzburg–Landau Equation with Multiresonant Forcing

Conway¹,

Riecke

2009

SIAM J. Appl. Dyn. Syst.

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Motivated by the rich variety of complex patterns observed on the surface of fluid layers that are vibrated at multiple frequencies, we investigate the effect of such resonant forcing on systems undergoing a Hopf bifurcation to spatially homogeneous oscillations. We use an extension of the complex Ginzburg-Landau equation that systematically captures weak forcing functions with a spectrum consisting of frequencies close to the 1:1-, the 1:2-, and the 1:3-resonance. By slowly modulating the amplitude of the 1:2-forcing component we render the bifurcation to subharmonic patterns supercritical despite the quadratic interaction introduced by the 1:3-forcing. Our weakly nonlinear analysis shows that quite generally the forcing function can be tuned such that resonant triad interactions with weakly damped modes stabilize subharmonic patterns comprised of four or five Fourier modes, which are similar to quasi-patterns with 4-fold and 5-fold rotational symmetry, respectively. Using direct simulations of the extended complex Ginzburg-Landau equation we confirm our weakly nonlinear analysis. In simulations domains of different complex patterns compete with each other on a slow time scale. As expected from energy arguments, with increasing strength of the triad interaction the more complex patterns eventually win out against the simpler patterns. We characterize these ordering dynamics using the spectral entropy of the patterns. For system parameters reported for experiments on the oscillatory Belousov-Zhabotinsky reaction we explicitly show that the forcing parameters can be tuned such that 4-mode patterns are the preferred patterns.

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“…Binks et al have shown experimentally that the depth of the layer [4] and the excitation frequency [5] affect significantly the regions, where either rolls or hexagons or squares are observed. In [6,7,8,9] it is revealed that a two-frequency excitation leads to a great variety of wave patterns. In particular, the observed patterns were triangles [6], superlattices formed by small and large hexagons [7], squares [6,7,8,9], and rhomboid pattern [9].…”

Section: Introductionmentioning

confidence: 99%

Faraday instability on viscous ferrofluids in a horizontal magnetic field: Oblique rolls of arbitrary orientation

Mekhonoshin

Lange

2002

Phys. Rev. E

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A linear stability analysis of the free surface of a horizontally unbounded ferrofluid layer of arbitrary depth subjected to vertical vibrations and a horizontal magnetic field is performed. A nonmonotonic dependence of the stability threshold on the magnetic field is found at high frequencies of the vibrations. The reasons of the decrease of the critical acceleration amplitude caused by a horizontal magnetic field are discussed. It is revealed that the magnetic field can be used to select the first unstable pattern of Faraday waves. In particular, a rhombic pattern as a superposition of two different oblique rolls can occur. A scaling law is presented which maps all data into one graph for the tested range of viscosities, frequencies, magnetic fields and layer thicknesses.

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Two-Mode Rhomboidal States in Driven Surface Waves

Cited by 60 publications

References 26 publications

Design of Parametrically Forced Patterns and Quasipatterns

Design of Parametrically Forced Patterns and Quasipatterns

Superlattice Patterns in the Complex Ginzburg–Landau Equation with Multiresonant Forcing

Faraday instability on viscous ferrofluids in a horizontal magnetic field: Oblique rolls of arbitrary orientation

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