Coupled instabilities drive quasiperiodic order-disorder transitions in Faraday waves Valeri Frumkin Department of Mathematics Massachusetts Institute of Technology

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Coupled instabilities drive quasiperiodic order-disorder transitions in Faraday waves

Valeri Frumkin∗

Department of Mathematics, Massachusetts Institute of Technology

Shreyas Gokhale†

Department of Physics, Massachusetts Institute of Technology

(Dated:)

We present an experimental study of quasiperiodic transitions between a highly ordered square-

lattice pattern and a disordered, defect-riddled state, in a circular Faraday system. We show that

the transition is driven initially by a long-wave amplitude modulation instability, which excites the

oscillatory transition phase instability, leading to the formation of dislocations in the Faraday lattice.

The appearance of dislocations damps amplitude modulations, which prevents further defects from

being created and allows the system to relax back to its ordered state. The process then repeats

itself in a quasiperiodic manner. Our experiments reveal a surprising coupling between two distinct

instabilities in the Faraday system, and suggest that such coupling may provide a generic mechanism

for quasiperiodicity in nonlinear driven dissipative systems.

When a thin layer of ﬂuid is subjected to uniform ver-

tical vibration with suﬃciently large amplitude, the ini-

tially ﬂat ﬂuid surface destabilizes to an ordered pattern

of sub-harmonic standing waves, known as Faraday waves

[1]. The Faraday system has been the subject of numer-

ous theoretical [2–4] and experimental [5–7] studies, and

it serves as a canonical example of a nonlinear pattern-

forming system [8, 9]. Its importance, however, goes be-

yond the study of pattern formation, as it manifests in

a wide range of physical systems across multiple length

scales. Faraday waves have been observed in systems

as disparate as Bose-Einstein condensates [10], soft elas-

tic solids [11], and even bodies of vibrated living earth-

worms [12]. In pilot-wave hydrodynamics, locally excited

Faraday waves store information about the trajectories

of walking droplets [13–15], while in hydrodynamic su-

perradiance they serve as the underlying mechanism for

sinusoidal oscillations of the droplet emission rate [16].

Since the Faraday system is readily accessible in the

lab, it allows for a detailed study of the complex transi-

tion from order to disorder in pattern-forming systems.

Speciﬁcally, when the driving amplitude is increased well

beyond the Faraday threshold, defects appear in the or-

dered Faraday lattice, leading to the emergence of spa-

tial disorder through a process that came to be known

as “defect-mediated turbulence” [17]. Defect formation

typically occurs via secondary instabilities, such as trans-

verse amplitude modulation (TAM) instability [18–20],

and the oscillatory transition phase (OTP) instability

[21]. In the former, the square Faraday pattern is mod-

ulated by long wavelength oscillations normal to the air-

ﬂuid interface, leading to an eventual loss of long-range

order with increasing driving amplitude. In the latter,

spatially uncorrelated elastic waves are excited within the

plane of the Faraday lattice, leading to the emergence of

defects. In both cases, as the defects are formed, the

square Faraday pattern exhibits a state of spatial inter-

mittency where the ordered and disordered phases can

coexist. With further increase in driving amplitude the

pattern loses any long range order and “melts” into a

fully spatiotemporally disordered state [22, 23].

A less known, but intriguing phenomenon is that of

temporal intermittency in the order-disorder transition in

the Faraday system. This phenomenon was ﬁrst reported

by Ezerskii [24], who observed that when the depth of the

liquid was chosen such that the group velocity of capillary

waves was close to the velocity of low-frequency gravity

modes, C=√ρg, resonant conditions would occur al-

lowing for eﬃcient energy transfer between the two. As

a result, in a speciﬁc parameter regime, weakly damped

gravity modes would get excited and slowly grow in am-

plitude, leading to an accelerated generation of higher

harmonics and a rapid transition to chaos. The system

would then alternate quasi-periodically between the low

frequency oscillations and fully disordered high frequency

modes. This behaviour was independent of the system’s

geometry, and the only condition for its emergence was

the aforementioned resonance between capillary waves

and the weakly damped gravity modes.

Here we describe a diﬀerent path to quasiperiodic dy-

namics in the Faraday system. We show that in the case

of a circular bath, for speciﬁc values of the bath radius,

amplitude modulations in the shape of vibrational modes

of a circular elastic membrane are excited in the Fara-

day lattice. These modes are resonantly ampliﬁed by the

driving, leading to the formation of dislocations via the

OTP instability. The increased dislocation density leads

to a rapid decay of spatial correlations, preventing forma-

tion of any additional dislocations, and allowing the sys-

tem to relax back to its ordered state. The process then

repeats itself. The phenomenon described here reveals a

surprising coupling between two distinct instabilities in

the Faraday system, namely, TAM and OTP.

Our experimental system consisted of a circular bath,

190 mm in diameter, that contained a 5 mm deep cir-

cular opening, with a diameter of 156 mm. The bath

arXiv:2210.10881v1 [physics.flu-dyn] 19 Oct 2022

FIG. 1. Quasiperiodic transitions between order and disorder in Faraday waves. (a-c) Snapshots of our system for

a ﬁxed driving frequency fd= 88 Hz showing an ordered Faraday wave lattice at peak vibration acceleration γ= 5.4g(a), an

intermittent partially disordered state at γ= 5.95g(b), and a chaotic state at γ= 6.5g. (d) A time sequence of snapshots

during a typical quasiperiod from the same data set as in (b), showing the onset of disorder followed by the clearing of defects

and reordering of the lattice.

was ﬁlled with silicon oil so that the resulting oil depth

was 5.6±0.2 mm above the inner opening, and ≈0.6

mm in the surrounding shallow layer. The shallow layer

acted as a wave damper and eliminated any eﬀects due to

sloshing of oil against the boundaries of the system. The

silicon oil had surface tension σ= 0.0209 N/m, viscos-

ity ν= 20 cSt, and density ρ= 0.965 ×103kg/m3. The

bath was vibrated vertically by an electromagnetic shaker

with forcing F(t) = γcos(2πfdt), with fdand γbeing the

frequency and peak vibrational acceleration, respectively.

We ensured spatial uniformity of the bath vibration by

connecting the shaker to the bath via a steel rod coupled

to a linear air bearing [25]. We monitored the vibrational

forcing using two accelerometers that were placed on op-

posite sides of the bath, ensuring a constant vibrational

acceleration amplitude to within ±0.002 g. To image the

emergent surface-wave pattern, we used a semi-reﬂective

mirror that was positioned at 45°between the bath and a

charge-coupled device (CCD) camera that was mounted

directly above the setup. The bath was illuminated with

a diﬀuse-light lamp facing the mirror horizontally, yield-

ing images with bright regions corresponding to horizon-

tal parts of the surface, speciﬁcally extrema or saddle

points. Before each experiment, we waited for 20 min-

utes for the system to stabilize and captured videos for

at least 10 mins, at a frame rate of fd/4 fps.

Fig.1 describes the typical evolution of the Faraday

system with increasing driving amplitude γ. For γ

slightly above the pattern-forming threshold, the Faraday

pattern takes the form of a square lattice characterized

by highly coherent long-range order (Fig.1a). With fur-

ther increase in the driving amplitude, line dislocations

appear in the lattice leading to a regime of coexistence

between ordered and disordered regions, reminiscent of

the intermittency route to chaos (Fig. 1b). The dislo-

cation density increases with an increase in the driving

amplitude, until ﬁnally the lattice “melts” into a fully

disordered, chaotic state (Fig. 1c).

The behavior of the system studied here is in stark

contrast with the typical intermittency route to chaos.

Speciﬁcally, there appears to be a small parameter range

γ1< γ < γ2, with γ1above the pattern-forming threshold

and γ2below the dislocation-forming threshold, where

amplitude modulations in the form of low frequency grav-

ity waves are excited. These secondary waves resonate

with the driving frequency and grow in amplitude over-

time, leading to the formation of dislocations in the Fara-

day lattice. As the number of dislocations increases,

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Coupledinstabilitiesdrivequasiperiodicorder-disordertransitionsinFaradaywavesValeriFrumkinDepartmentofMathematics,MassachusettsInstituteofTechnologyShreyasGokhaleyDepartmentofPhysics,MassachusettsInstituteofTechnology(Dated:)Wepresentanexperimentalstudyofquasiperiodictransitionsbetweenahighlyordere...

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Coupled instabilities drive quasiperiodic order-disorder transitions in Faraday waves Valeri Frumkin Department of Mathematics Massachusetts Institute of Technology

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