Zigzag Instability Research Articles

Zigzag instability of columnar Taylor–Green vortices in a strongly stratified fluid

We investigate the dynamics of a columnar Taylor–Green vortex array under strong stratification, focusing on Froude numbers $0.125\leq Fr \leq 1.0$ , with the aim of identifying and understanding the primary instabilities that lead to the vortices’ breakdown. Linear stability analysis reveals that the fastest-growing vertical wavenumber scales with $Fr^{-1}$ , while the dimensionless growth rate remains approximately constant. The most unstable eigenmode, identified as the mixed hyperbolic mode by Hattori et al. (J. Fluid Mech., vol. 909, 2021, A4), bears significant similarities to the zigzag instability, first discovered by Billant & Chomaz (J. Fluid Mech., vol. 418, 2000, pp. 167–188). Direct numerical simulations further confirm that the zigzag instability is crucial in amplifying initial random perturbations to finite amplitude, with the flow structure and modal growth rate consistent with the linear stability analysis. In particular, the characteristic vertical length scale of turbulence matches that of the fastest-growing linear mode. These findings underscore the broader relevance of the zigzag instability mechanism beyond its initial discovery in vortex pairs, demonstrating its role in facilitating direct energy transfer from vertically uniform vortical motions to a characteristic vertical length scale proportional to $Fr$ in strongly stratified flows.

Confinement and magnetic-field effect on chiral ferroelectric nematic liquid crystals in Grandjean-Cano wedge cells.

We explore the structure and magnetic-field response of edge dislocations in Grandjean-Cano wedge cells filled with chiral mixtures of the ferroelectric nematic mesogen DIO. Upon cooling, the ordering changes from paraelectric in the cholesteric phase N^{*} to antiferroelectric in the smectic SmZ_{A}^{*} and to ferroelectric in the cholesteric N_{F}^{*}. Dislocations of the Burgers vector b equal to the helicoidal pitch P are stable in all three phases, while dislocations with b=P/2 exist only in the N^{*} and SmZ_{A}^{*}. The b=P/2 dislocations split into pairs of τ^{-1/2}λ^{+1/2} disclinations, while the thick dislocations b=P are pairs of nonsingular λ^{-1/2}λ^{+1/2} disclinations. The polar order makes the τ^{-1/2} disclinations unstable in the N_{F}^{*} phase, as they should be connected to singular walls in the polarization field. We propose a model of transformation of the composite τ^{-1/2} line-wall defect into a nonsingular λ^{-1/2} disclination, which is paired up with a λ^{+1/2} line to form a b=P dislocation. The SmZ_{A}^{*} behavior in the in-plane magnetic field is different from that of the N_{F}^{*} and N^{*}: the dislocations show no zigzag instability, and the pitch remains unchanged in the magnetic fields up to 1 T. The behavior is associated with the finite compressibility of smectic layers.

Ring structural transitions in strongly coupled dusty plasmas.

This paper presents a numerical study of ring structural transitions in strongly coupled dusty plasma confined in a ring-shaped (quartic) potential well with a central barrier, whose axis of symmetry is parallel to the gravitational attraction. It is observed that increasing the amplitude of the potential leads to a transition from a ring monolayer structure (rings of different diameters nested within the same plane) to a cylindrical shell structure (rings of similar diameter aligned in parallel planes). In the cylindrical shell state, the ring's alignment in the vertical plane exhibits hexagonal symmetry. The ring transition is reversible, but exhibits hysteresis in the initial and final particle positions. As the critical conditions for the transitions are approached, the transitional structure states exhibit zigzag instabilities or asymmetries on the ring alignment. Furthermore, for a fixed amplitude of the quartic potential that results in a cylinder-shaped shell structure, we show that additional rings in the cylindrical shell structure can be formed by decreasing the curvature of the parabolic potential well, whose axis of symmetry is perpendicular to the gravitational force, increasing the number density, and lowering the screening parameter. Finally, we discuss the application of these findings to dusty plasma experiments with ring electrodes and weak magnetic fields.

On the bandwidth of stable nonlinear stripe patterns in finite size systems.

Nonlinear stripe patterns occur in many different systems, from the small scales of biological cells to geological scales as cloud patterns. They all share the universal property of being stable at different wavenumbers q, i.e., they are multistable. The stable wavenumber range of the stripe patterns, which is limited by the Eckhaus- and zigzag instabilities even in finite systems for several boundary conditions, increases with decreasing system size. This enlargement comes about because suppressing degrees of freedom from the two instabilities goes along with the system reduction, and the enlargement depends on the boundary conditions, as we show analytically and numerically with the generic Swift-Hohenberg (SH) model and the universal Newell-Whitehead-Segel equation. We also describe how, in very small system sizes, any periodic pattern that emerges from the basic state is simultaneously stable in certain parameter ranges, which is especially important for the Turing pattern in cells. In addition, we explain why below a certain system width, stripe patterns behave quasi-one-dimensional in two-dimensional systems. Furthermore, we show with numerical simulations of the SH model in medium-sized rectangular domains how unstable stripe patterns evolve via the zigzag instability differently into stable patterns for different combinations of boundary conditions.

Zigzag instability of biased pusher swimmers

Microorganisms self-propelling in a fluid create flow fields that impact the dynamics of other swimmers. Some organisms can be biased and have a preferred swimming direction, e.g., those displaying gyrotaxis, chemotaxis or phototaxis, and as a result often focus along thin lines. Here we use numerical computations and far-field theoretical calculations to show that the position of a collection of biased swimmers moving along a line is unstable to a zigzag mode when the swimmers act on the fluid as pusher dipoles. This instability takes the form of periodic transverse oscillations in the position of the swimmers. We predict theoretically that the most unstable wavelength is equal to twice the inter-swimmer distance and that the growth rate of the instability increases linearly with the magnitude of the stresslet, both of which are in quantitative agreement with our numerical simulations.

Pattern Dynamics in a Diffusive Gierer–Meinhardt Model

The purpose of the present paper is to investigate the pattern formation and secondary instabilities, including Eckhaus instability and zigzag instability, of an activator–inhibitor system, known as the Gierer–Meinhardt model. Conditions on the Hopf bifurcation and the Turing instability are obtained through linear stability analysis at the unique positive equilibrium. Then, the method of weakly nonlinear analysis is used to derive the amplitude equations. Especially, by adding a small disturbance to the Turing instability critical wave number, the spatiotemporal Newell–Whitehead–Segel equation of the stripe pattern is established. It is found that Eckhaus instability and zigzag instability may occur under certain conditions. Finally, Turing and non-Turing patterns are obtained via numerical simulations, including spotted patterns, mixed patterns, Eckhaus patterns, spatiotemporal chaos, nonconstant steady state solutions, spatially homogeneous periodic solutions and spatially inhomogeneous solutions in two-dimensional or one-dimensional space. Theoretical analysis and numerical results are in good agreement for this diffusive Gierer–Meinhardt model.

Orientation and morphology of solid-state dewetting holes

We report on the orientation and morphology of solid-state dewetting holes obtained from a kinetic Monte Carlo model with nearest neighbor and next nearest neighbor interactions on a cubic lattice. The morphologies found in simulations share strong similarities with those of diffusion-limited crystal growth: We find compact shapes, isotropic and anisotropic seaweed shapes, and dendritic shapes. Some of these shapes have been observed in solid-state dewetting experiments with various semi-conductors or metals on insulating substrates. The sides of the fingers can exhibit two types of zigzag instabilities with ${90}^{\ensuremath{\circ}}$ or ${45}^{\ensuremath{\circ}}$. Due to fourfold symmetry, dewetting hole shapes are found to exhibit two possible orientations with fingers along the (110) or (100) directions. We find that the rotation transition from one orientation to the other can be observed in our model by varying anisotropy, temperature, or dewetting strength. The two orientations correspond, respectively, to experimental observations of dewetting for silicon and germanium on insulator and to the dewetting of metal films with different reducing gas flow rate.

Stripes on finite domains: Why the zigzag instability is only a partial story.

Stationary periodic patterns are widespread in natural sciences, ranging from nano-scale electrochemical and amphiphilic systems to mesoscale fluid, chemical, and biological media and to macro-scale vegetation and cloud patterns. Their formation is usually due to a primary symmetry breaking of a uniform state to stripes, often followed by secondary instabilities to form zigzag and labyrinthine patterns. These secondary instabilities are well studied under idealized conditions of an infinite domain; however, on finite domains, the situation is more subtle since the unstable modes depend also on boundary conditions. Using two prototypical models, the Swift-Hohenberg equation and the forced complex Ginzburg-Landau equation, we consider finite size domains with no flux boundary conditions transversal to the stripes and reveal a distinct mixed-mode instability that lies in between the classical zigzag and the Eckhaus lines. This explains the stability of stripes in the mildly zigzag unstable regime and, after crossing the mixed-mode line, the evolution of zigzag stripes in the bulk of the domain and the formation of defects near the boundaries. The results are of particular importance for problems with large timescale separation, such as bulk-heterojunction deformations in organic photovoltaic and vegetation in semi-arid regions, where early temporal transients may play an important role.

Nonlinear magneto convection due to horizontal magnetic field and vertical axis of rotation due to thermal and compositional buoyancy

Abstract This paper presents linear and weakly non linear analysis of magnetoconvection due to horizontal magnetic field and vertical axis of rotation due to thermal and compositional buoyancy. For linear stability analysis, the normal mode analysis is utilized to find the Rayleigh number which is the function of Rosby number, Lewis number and Ekman number. Also, the correlation between Rayleigh number and wave number is graphically analyzed. The Newell-Whitehead scheme is employed to derive two dimensional amplitude equation. The conditions for Eckhaus and Zigzag instabilities are determined. The system of coupled Landau-Ginzburg equations are derived. For Standing waves and Travelling waves, the stability regions are identified.

Eckhaus and zigzag instability in a chemotaxis model of multiple sclerosis

We present a theoretical and numerical study of the bifurcations of the stationary patterns supported by a chemotactic model of Multiple Sclerosis (MS). We derive the normal forms of the dynamics which allows to predict the appearance and stabilization of the emerging branches describing the concentric patterns typical of Balo's sclerosis, a very aggressive variant of MS. Spatial modulation of the Turing-type structures through a zigzag instability is also addressed. The nonlinear stage of the Eckhaus and zigzag instability is investigated numerically: defect-mediated wavenumber adjustments are recovered and the time of occurrence of phase-slips is studied as the system parameters are varied.

Stabilizing a homoclinic stripe

For a large class of reaction-diffusion systems with large diffusivity ratio, it is well known that a two-dimensional stripe (whose cross-section is a one-dimensional homoclinic spike) is unstable and breaks up into spots. Here, we study two effects that can stabilize such a homoclinic stripe. First, we consider the addition of anisotropy to the model. For the Schnakenberg model, we show that (an infinite) stripe can be stabilized if the fast-diffusing variable (substrate) is sufficiently anisotropic. Two types of instability thresholds are derived: zigzag (or bending) and break-up instabilities. The instability boundaries subdivide parameter space into three distinct zones: stable stripe, unstable stripe due to bending and unstable due to break-up instability. Numerical experiments indicate that the break-up instability is supercritical leading to a 'spotted-stripe' solution. Finally, we perform a similar analysis for the Klausmeier model of vegetation patterns on a steep hill, and examine transition from spots to stripes.This article is part of the theme issue 'Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (part 2)'.

Effect of growth rate on the microstructural transition and microhardness of directionally solidified Ni–11.8 wt% Si hypereutectic alloy

Ni–11.8 wt% Si hypereutectic alloys were directionally solidified under growth rates of 0.5–15 μm/s. The microstructures of Ni–11.8 wt% Si alloys were characterized by field-emission scanning electron microscopy–energy dispersive spectrometry, and the phase composition was identified by X-ray diffraction. The alloys exhibited three types of microstructures: rods, wavy lamellae, and straight lamellae. The alloys grown at 4–12 μm/s exhibited a completely lamellar structure that consisted of α-Ni and β-Ni3Si phases. During lamellar–rod transition, coarse lamellae first underwent zigzag instability to form wavy lamellae and then transformed into rods. An interface deflection parameter was defined based on the deviation of the interface shape from a flat interface and reached the maximum value under maximum lamellar spacing. Moreover, zigzag instability resulted in a macroscopically flat liquid-solid interface. Out of the three microstructures, the rod-like structure exhibited the highest microhardness, followed by straight lamellae and then wavy lamellae. The difference in microhardness was related to the volume fraction of α-Ni phase and microstructural type. The lamellar microstructure more effectively improved the microhardness of eutectic materials than the rod-like microstructure.

Morphological instability of lamellar structures in directionally solidified Ni–Ni3Si alloys

The morphological instability of lamellar structures in Ni–Ni3Si eutectic and hypereutectic alloys directionally solidified at low growth rates was investigated. The first instability in large lamellar structures was zigzag instability, which formed curved lamellae. A zigzag pattern was first displayed in three dimensions. The diffusion-limited growth of the Ni3Si phase decreased phase width and spacing, consequently causing zigzag instability. The reduced spacing was observed at λ/λave = 0.9. After zigzag instability, the microstructure of the eutectic alloy turned into a labyrinth structure and lamellar fragmentation. However, in hypereutectic alloys, shape transition from lamellae to rods occurred, in turn, by the broken lamellae or elongated rods to dumbbell-shaped rods, peanut-shaped rods, and circular rods.

Instabilities of convection patterns in a shear-thinning fluid between plates of finite conductivity.

Rayleigh-Bénard convection in a horizontal layer of a non-Newtonian fluid between slabs of arbitrary thickness and finite thermal conductivity is considered. The first part of the paper deals with the primary bifurcation and the relative stability of convective patterns at threshold. Weakly nonlinear analysis combined with Stuart-Landau equation is used. The competition between squares and rolls, as a function of the shear-thinning degree of the fluid, the slabs' thickness, and the ratio of the thermal conductivity of the slabs to that of the fluid is investigated. Computations of heat transfer coefficients are in agreement with the maximum heat transfer principle. The second part of the paper concerns the stability of the convective patterns toward spatial perturbations and the determination of the band width of the stable wave number in the neighborhood of the critical Rayleigh number. The approach used is based on the Ginzburg-Landau equations. The study of rolls stability shows that: (i) for low shear-thinning effects, the band of stable wave numbers is bounded by zigzag instability and cross-roll instability. Furthermore, the marginal cross-roll stability boundary enlarges with increasing shear-thinning properties; (ii) for high shear-thinning effects, Eckhaus instability becomes more dangerous than cross-roll instability. For square patterns, the wave number selection is always restricted by zigzag instability and by "rectangular Eckhaus" instability. In addition, the width of the stable wave number decreases with increasing shear-thinning effects. Numerical simulations of the planform evolution are also presented to illustrate the different instabilities considered in the paper.

Effects of second diffusing component and cross diffusion on primary and secondary thermoconvective instabilities in couple stress liquids

Linear and weakly nonlinear analyses are made for the Rayleigh-Benard convection in two-component couple-stress liquids with the Soret effect. Conditions for pitchfork, Hopf, Takens-Bogdanov, and codimension-two bifurcations are presented. The Lorenz model is used to study the inverted bifurcation. Positive values of the Soret coefficient favor a pitchfork bifurcation, whereas negative values favor a Hopf bifurcation. Takens-Bogdanov and codimension-two bifurcations are not as much influenced by the Soret coefficient as pitchfork and Hopf bifurcations. The influence of the Soret coefficient on the inverted bifurcation is similar to the influence on the pitchfork bifurcation. The influence of other parameters on the aforementioned bifurcations is also similar as reported earlier in the literature. Using the Newell-Whitehead-Segel equation, the condition for occurrence of Eckhaus and zigzag secondary instabilities is obtained. The domain of appearance of Eckhaus and zigzag instabilities expands due to the presence of the Soret coefficient for positive values. The Soret coefficient with negative values enhances heat transport, while positive values diminish it in comparison with heat transport for the case without the Soret effect. The dual nature of other parameters in influencing heat and mass transport is shown by considering positive and negative values of the Soret coefficient.

On the effect of conducting lid for natural convection in a horizontal fluid layer

In this study, effect of conducting lid attached in the horizontal fluid layer for Rayleigh-Bénard natural convection was discussed. Geometry which was taken into account was horizontal layer of fluid with conducting lid at the bottom. Conducting lid was heated from the bottom wall and the thermal energy was transferred to fluid layer through the solid/fluid interface, and then was cooled down from above. Thermal conductivity ratio varied to investigate the thermal behavior at the solid/fluid interface. Periodic boundary conditions were employed in the horizontal direction to allow lateral freedom for the convection cells. Prior to investigate thermal behavior of fluid layer with conducting lid, pure Rayleigh-Bénard natural convection without conducting body was reproduced. From two-dimensional Rayleigh-Bénard natural convection, an extension of wavelength of circulating roll cell was examined. A two-dimensional solution was obtained using Chebyshev spectral multi-domain methodology for different Rayleigh number at which the thermal behavior was evolved from steady state to chaotic pattern. Three-dimensional pure Rayleigh-Bénard natural convection was calculated from relatively low Rayleigh number such as 4×103 in order to investigate an evolution of thermal plume undergoing a representative zig-zag instability. In order to compare thermal behavior of fluid layer between with- and without conducting lid, effective Rayleigh numbers, Raeff, was introduced and 106 was applied. A solid lid at the bottom affects the flow pattern in that the flow is restricted to increase the dimensionless thermal conductivity. For three-dimensional simulation, periodic boundary condition was also applied along the span-wise direction which was discretized through a Fourier serious expansion with a uniform mesh configuration. For high effective Rayleigh number, thermal flow field was captured by visualizing the three-dimensional vortical structure. The flow behavior at the provided effective Rayleigh number showed a coherent pattern regardless of the magnitude of the thermal conductivity ratio.

Existence, Stability, and Dynamics of Ring and Near-Ring Solutions to the Saturated Gierer--Meinhardt Model in the Semistrong Regime

We analyze a singularly perturbed reaction-diffusion system in the semistrong diffusion regime in two spatial dimensions where an activator species is localized to a closed curve, while the inhibitor species exhibits long-range behavior over the domain. In the limit of small activator diffusivity, we derive a new moving boundary problem characterizing the slow time evolution of the curve, which is defined in terms of a quasi-steady state inhibitor diffusion field and its properties on the curve. Numerical results from this curve evolution problem are illustrated for the Gierer--Meinhardt model with saturation (GMS) in the activator kinetics. A detailed analysis of the existence, stability, and dynamics of ring and near-ring solutions for the GMS model is given, whereby the activator concentrates on a thin ring concentric within a circular domain. A key new result for this ring geometry is that by including activator saturation there is a qualitative change in the phase portrait of ring equilibria, in that there is an S-shaped bifurcation diagram for ring equilibria, which allows for hysteresis behavior. In contrast, without saturation, it is well known that there is a saddle-node bifurcation for the ring equilibria. For a near-circular ring, we develop an asymptotic expansion up to quadratic order to fully characterize the normal velocity perturbations from our curve evolution problem. In addition, we also analyze the linear stability of the ring solution to both breakup instabilities, leading to the disintegration of a ring into localized spots, and zig-zag instabilities, leading to the slow shape deformation of the ring. We show from a nonlocal eigenvalue problem that activator saturation can stabilize breakup patterns that otherwise would be unstable. Through a detailed matched asymptotic analysis, we derive a new explicit formula for the small eigenvalues associated with zig-zag instabilities, and we show that they are equivalent to the velocity perturbations induced by the near-circular ring geometry. Finally, we present full numerical simulations from the GMS PDE system that confirm the predictions of the analysis.

Layers and internal waves in uniformly stratified fluids stirred by vertical grids

Laboratory experiments in which uniformly stratified fluids are stirred by horizontally moving vertical grids, or arrays of vertical rods, are reviewed to examine their consistency and to compare their findings, particularly those relating to the generation of layers. Selected experiments are of three types, those in which (a) turbulence spreads from a horizontally confined region where it is continuously generated by an oscillating grid; (b) grid stirring is maintained throughout a rectangular tank; or (c) a ‘cloud’ of turbulence decays after a short period of horizontally localized grid mixing. In all the experiments turbulence is generated over the full vertical extent of the tank. In the experiments of types (a) and (c) layers of comparable scale are observed to intrude into the ambient fluid from the turbulent region. In the type (b) experiments, layers form only when the time interval between the passage of the grid through the stratified fluid is sufficiently long, allowing turbulence to decay substantially between grid strokes. Two mechanisms are found to be dominant in the production of layers. In experiments of type (a) and (c) overturning eddies in the turbulent region of scale significantly larger than the Ozmidov length scale collapse and spread, intruding and forming layers in the adjoining laminar region. Internal shear waves propagating ahead of the intrusions have a vertical wavelength that is approximately twice the layer height. In type (b) experiments, layers are formed through a process described by Holford & Linden (Dyn. Atmos. Oceans, vol. 30, 1999a, pp. 173–198): the bending of vortices shed by the grid bars. The height of the layers is approximately half the vertical wavelength of internal shear waves that travel at the speed of the grid. It is proposed that the flow field of the shear waves bends the vortices, resulting in diapycnal mixing that forms the layers. The relationship of layers and internal shear waves in the experiments is therefore as follows: in type (a) and (c) experiments internal waves are generated by layers intruding from the turbulent region into the quiescent stratified region, but in experiments of type (b) internal waves drive and dictate the height of the layers; layers are generated as a consequence of vortex bending by internal waves. There is insufficient evidence to establish whether zigzag instability, either of vortex pairs or of vortex streets shed by a moving grid, accounts for the layers in any of the three types of grid experiments. The Phillips and Posmentier instability may reinforce layers formed by other processes. The generation of pancake vortices or vortical mode motion is left for later review.

Short-wave vortex instability in stratified flow

In this paper we investigate a new instability of the Lamb–Chaplygin dipole in a stratified fluid. Through numerical linear stability analysis, a secondary peak in the growth rate emerges at vertical scales about an order of magnitude smaller than the buoyancy scale Lb=U/N where U is the characteristic velocity and N is the Brunt–Väisälä frequency. This new instability exhibits a growth rate that is similar to, and even exceeds, that of the zigzag instability, which has the characteristic length of the buoyancy scale. This instability is investigated for a wide range of Reynolds numbers, Re=2000–20000, and horizontal Froude numbers, Fh=0.05–0.2, where Fh=U/NR, Re=UR/ν, R is the radius of the dipole, and ν is the kinematic viscosity. A range of vertical scales is explored from above the buoyancy scale to the viscous damping scale. Additionally, evidence is presented that the length scale and growth rate of this new instability are partially determined by the buoyancy Reynolds number, Reb=Fh2Re.

Pattern formation in the FitzHugh–Nagumo model

In this paper, we investigate the effect of diffusion on pattern formation in FitzHugh–Nagumo model. Through the linear stability analysis of local equilibrium we obtain the condition how the Turing bifurcation, Hopf bifurcation and the oscillatory instability boundaries arise. By using the method of the weak nonlinear multiple scales analysis and Taylor series expansion, we derive the amplitude equations of the stationary patterns. The analysis of amplitude equations shows the occurrence of different complex phenomena, including Turing instability Eckhaus instability and zigzag instability. In addition, we apply this analysis to FitzHugh–Nagumo model and find that this model has very rich dynamical behaviors, such as spotted, stripe and hexagon patterns. Finally, the numerical simulation shows that the analytical results agree with numerical simulation.