Figure 10 - uploaded by Edgar Knobloch
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(Colour online) Bifurcation diagrams for even-parity two-pulse states in a Γ = 14 domain. (a) Solutions based on 12 rolls. (b) Solutions based on 14 rolls. Solid dots indicate the location of the solutions shown in figure 11.
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Binary fluid mixtures with a negative separation ratio heated from below exhibit steady spatially localized states called convectons for supercritical Rayleigh numbers. Numerical continuation is used to compute such states in the presence of both Neumann boundary conditions and no-slip no-flux boundary conditions in the horizontal. In addition to t...
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... figure 10, we show the effect of this change on even-parity two-pulse states. Figure 10(a) shows two-pulse states composed of identical even-parity convectons. ...
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... figure 10, we show the effect of this change on even-parity two-pulse states. Figure 10(a) shows two-pulse states composed of identical even-parity convectons. The branch labelled 2PC 12 is of type (0, 0) and consists of an even-parity central convecton with a rising warm plume in the centre and a pair of wall-attached states at either end of the domain ( figure 11a, top). ...
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... together back-to-back, the latter represent another even convecton with a rising plume in the centre. Thus, the observed structure is in fact identical to the (0, 0) two-pulse configuration that one are not symmetry-related and hence generically lie on distinct solution branches, as seen in figure 10(b). The question now arises as to the presence of two-pulse states with overall odd parity, i.e. with overall point symmetry. ...
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... is a consequence of their mutual interaction. Indeed, in this example the state obtained by reflecting it in the boundary x = Γ /2 differs from a Γ /2 translation, in contrast to the situation in figure 10, i.e. the state 2PH 13 , like 2PC 13 , does not satisfy PBC with period Γ . Moreover, since the 2PH 13 state is not a bound state of two pure parity states, it changes its appearance along the branch. ...
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... in the odd-parity case the transition region between the two-pulse and hole states is characterized by markedly asymmetric growth of the convectons once these reach close to the lateral walls. In this case, the two-pulse branch in figure 14(b) is of H-type and very similar to the branch 2PH 12 in figure 10(a). Indeed, the solutions in figure 16(a, top) and figure 11(a, bottom) are very similar. ...
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Citations
... Despite the challenge of analyzing and simulating mathematical equations in three spatial dimensions, localized patterns have been identified in numerous settings. This includes water waves [1,49,50,189], fluid convection [162,177,213,227], turbulence [13,193,211], and crystal formation in soft matter [217,222]. What further makes the analysis of localization in three spatial dimensions difficult is that it can manifest itself in one, two, or three directions. ...
... The study of convectons goes back to the work of Blanchflower [34] and has been predominantly undertaken in two models for fluid convection. The first of which is binary fluid convection, where two horizontal layers of miscible fluids are heated from below [177,[251][252][253], and the second is doubly diffusive convection, where they studied a two-component fluid subject to horizontal gradients of temperature and concentration [25,28,29,227] and with vertical gradients [24]. Various localized solutions can be found via direct numerical simulations Figure 46: An illustration of the 3D doubly diffusive convectons found in [26]. ...
... of the models, and numerical continuation can be used to further understand their bifurcation diagrams [24,26,32,[176][177][178]. In particular, localized solutions can form bound states called multiconvectons [177], which are similar to the multipulse solutions studied by Knobloch et al. [141]. ...
Localized patterns are coherent structures embedded in a quiescent state and occur in both discrete and continuous media across a wide range of applications. While it is well-understood how domain covering patterns (for example stripes and hexagons) emerge from a pattern-forming/Turing instability, analyzing the emergence of their localized counterparts remains a significant challenge. There has been considerable progress in studying localized patterns over the past few decades, often by employing innovative mathematical tools and techniques. In particular, the study of localized pattern formation has benefited greatly from numerical techniques; the continuing advancement in computational power has helped to both identify new types patterns and further our understanding of their behavior. We review recent advances regarding the complex behavior of localized patterns and the mathematical tools that have been developed to understand them, covering various topics from spatial dynamics, exponential asymptotics, and numerical methods. We observe that the mathematical understanding of localized patterns decreases as the spatial dimension increases, thus providing significant open problems that will form the basis for future investigations.
... Temporal complexity has also been found in various forms (Spina et al. 1998;Batiste et al. 2001) and is generated in a number of ways (Knobloch et al. 1986;Rucklidge 1992;Beaume 2020). Work focusing on steady state dynamics also revealed intricate phenomena like spatial localisation in the presence (Mercader et al. 2009(Mercader et al. , 2011 and in absence (Beaume et al. 2011) of Soret effect. Localised convection states, called convectons, are found on solution branches exhibiting oscillatory trajectories in parameter space in a behaviour called snaking (Knobloch 2015). ...
Doubly diffusive convection is considered in a vertical slot where horizontal temperature and solutal variations provide competing effects to the fluid density while allowing the existence of a conduction state. In this configuration, the linear stability of the conductive state is known, but the convection patterns arising from the primary instability have only been studied for specific parameter values. We have extended this by determining the nature of the primary bifurcation for all values of the Lewis and Prandtl numbers using a weakly nonlinear analysis. The resulting convection branches are extended using numerical continuation and we find large-amplitude steady convection states can coexist with the stable conduction state for sub- and supercritical primary bifurcations. The stability of the convection states is investigated and attracting travelling waves and periodic orbits are identified using time-stepping when these steady states are unstable.
... Doubly diffusive convection is noted for its ability to produce a variety of complex dynamics and patterns. In particular, this system has often been used to investigate localisation in fluids [2,6,7,11,12,34,35,36,37,47], where the spatially localised steady states typically take the form of convection rolls within a background conduction (motionless fluid) state and are referred to as convectons [13]. ...
... This process is referred to as homoclinic snaking [50] and results in the coexistence of a multiplicity of localised states for the same parameter values. The study of localised states in fluids has unsurprisingly revolved around these parameter values where subcriticality and bistability are observed [2,5,18,33,35,36,39,41,47] but these characteristics are not, in fact, necessary for spatial localisation. Recent studies on systems involving a conserved quantity, including magnetoconvection [16,18,32,33], rotating convection [5], vibrating granular or fluid layers [17,38], optics [20] and phase-field crystal models [44], have shown that localised states may lie on slanted snaking branches that can extend outside of the bistable region or may bifurcate from supercritical primary branches via modulational instabilities [5,18]. ...
Fluids subject to both thermal and compositional variations can undergo doubly diffusive convection when these properties both affect the fluid density and diffuse at different rates. A variety of patterns can arise from these buoyancy-driven flows, including spatially localised states known as convectons, which consist of convective fluid motion localised within a background of quiescent fluid. We consider these states in a vertical slot with the horizontal temperature and solutal gradients providing competing effects to the fluid density while allowing the existence of a conduction state. In this configuration, convectons have been studied with specific parameter values where the onset of convection is subcritical, and the states have been found to lie on a pair of secondary branches that undergo homoclinic snaking in a parameter regime below the onset of linear instability. In this paper, we show that convectons persist into parameter regimes in which the primary bifurcation is supercritical and there is no bistability, despite coexistence between the stable conduction state and large-amplitude convection. We detail this transition by considering spatial dynamics and observe how the structure of the secondary branches becomes increasingly complex owing to the increased role of inertia at low Prandtl numbers.
... The broad phenomenon of 'lower-order' states interacting to form new coherent structures has been seen in other physical systems. For example, the interaction of solitons in water waves (see, for example Drazin & Johnson 1989) and nonlinear optics (see, for example Akhmediev & Ankiewicz 2005), spatially localised states in convection systems (see, for example Mercader et al. 2010) and oscillons in granular particulate flow (see, for example Umbanhowar, Melo & Swinney 1996). A particular anomaly of our system is that we cannot smoothly move from a n-bubble state to a m-bubble state by continuation or branch-switching methods because the topologies of the systems are different. ...
We study the dynamics of two air bubbles driven by the motion of a suspending viscous fluid in a Hele-Shaw channel with a small elevation along its centreline via physical experiment and numerical simulation of a depth-averaged model. For a single-bubble system we establish that, in general, the bubble propagation speed monotonically increases with bubble volume so that two bubbles of different sizes, in the absence of any hydrodynamic interactions, will either coalesce or separate in a finite time. However, our experiments indicate that the bubbles interact and that an unstable two-bubble state is responsible for the eventual dynamical outcome: coalescence or separation. These results motivate us to develop an edge-tracking routine and to calculate these weakly unstable two-bubble steady states from the governing equations. The steady states consist of pairs of ‘aligned’ bubbles that appear on the same side of the centreline with the larger bubble leading. We also discover, through time-dependent simulations and physical experiment, another class of two-bubble states that, surprisingly, are stable. In contrast to the ‘aligned’ steady states, these bubbles appear on either side of the centreline and are ‘offset’ from each other. We calculate the bifurcation structures of both classes of steady states as the flow rate and bubble volume ratio are varied. We find that they exhibit intriguing similarities to the single-bubble bifurcation structure, which has implications for the existence of $n$ -bubble steady states.
... Time-dependent spatially localized states in dissipative systems, such as action potentials in physiology [1][2][3][4] and oscillons in Faraday waves [5][6][7], have attracted considerable interest in the past half-century [8,9]. Work in these fields not only provided insights into the original model (e.g., neural and cardiac) systems but also stimulated understanding of pattern formation mechanisms, which in turn proved applicable in other settings, such as in chemical reactions [10], nonlinear optics [11][12][13], fluid convection [14][15][16][17], and even sound discrimination in the inner ear [18]. Importantly, advances in pattern formation theory require the development of nonlinear methodologies and, specifically, numerical (continuation) methods for differential equations [9], such as the packages AUTO [19], MatCont [20], and pde2path [21]. ...
Oscillons, i.e., immobile spatially localized but temporally oscillating structures, are the subject of intense study since their discovery in Faraday wave experiments. However, oscillons can also disappear and reappear at a shifted spatial location, becoming jumping oscillons (JOs). We explain here the origin of this behavior in a three-variable reaction-diffusion system via numerical continuation and bifurcation theory, and show that JOs are created via a modulational instability of excitable traveling pulses (TPs). We also reveal the presence of bound states of JOs and TPs and patches of such states (including jumping periodic patterns) and determine their stability. This rich multiplicity of spatiotemporal states lends itself to information and storage handling.
... When the bifurcation to steady states is subcritical, secondary bifurcations have been found that lead to the formation of steady, spatially localized states known as convectons [19]. Their formation and their bifurcation diagram was characterized by Mercader et al. [20,21]: the convectons lie on a pair of branches undergoing well-bounded oscillations in parameter space in a behavior called snaking. Time-dependent localized states, in the form of traveling convection rolls within a spatially localized envelope traveling at a different speed, have also been observed to produce similar bifurcation diagrams [22] and the interplay between their stable and unstable manifolds has been elucidated by Watanabe et al. [23]. ...
... 12. Bifurcation diagram for system(21),(22) with β = 2 and γ = 0.1. The steady solutions are represented via the energy E = c + τ 2 versus the reduced Rayleigh number r. ...
Doubly diffusive convection driven by horizontal gradients of temperature and salinity is studied in a three-dimensional enclosure of square horizontal cross section and large aspect ratio. Previous studies focused on the primary instability and revealed the formation of subcritical branches of spatially localized states. These states lose stability because of their twist instability, thereby precluding the presence of any related stable steady states beyond the primary bifurcation and giving rise to spontaneous temporal complexity for supercritical parameter values. This paper investigates the emergence of this behavior. In particular, chaos is shown to be produced at a crisis bifurcation located close to the primary bifurcation. The critical exponent related to this crisis bifurcation is computed and explains the unusually abrupt transition. The construction of a low-dimensional model highlights that only a few requirements are necessary for this type of transition to occur. As a consequence, it is believed to be observable in many other systems.
... We have to point out that this scenario is not intrinsic to the field of nonlinear optics, but generic, appearing in many different context from hydrodynamics and material sciences to plant ecology [11,31,[38][39][40][41][42][43][44][45]. ...
... The homoclinic snaking phenomenon is not intrinsic to this system, but generic, arising in many different fields ranging from optics to plant ecology. [11,31,[38][39][40][41][42][43][44][45]. For the set of (∆ 1 , S)−parameters studied here, homoclinic snaking is well preserved. ...
We study the formation of localized patterns arising in doubly resonant dispersive optical parametric oscillators. They form through the locking of fronts connecting a continuous-wave and a Turing pattern state. This type of localized pattern can be seen as a slug of the pattern embedded in a homogeneous surrounding. They are organized in terms of a homoclinic snaking bifurcation structure, which is preserved under the modification of the control parameter of the system. We show that, in the presence of phase mismatch, localized patterns can undergo oscillatory instabilities which make them breathe in a complex manner.
... Laboratory experiments have observed convective states taking the form of traveling waves (TW) [2], localized TW (LTW) [3,4], stationary overturning convection (SOC) [2], blinking [5,6], counterpropagating waves (CPW, or "chevrons") [7,8], repeated transients [7,8] and dispersive chaos [9]. After these pioneering studies, many research by direct numerical simulations (DNS) were conducted to further understand and identify the mechanisms of the convective states, see e.g., References [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27]. By DNS one can get easily the values at any point of the flow fields and extract from them information difficult or impossible to obtain in the experiments, this allows a better understanding of the convective states. ...
... For low Rayleigh numbers, it is also challenging to investigate the problem by 3D simulations for a broad range of control parameters. Two-dimensional (2D) simulation, which is substantially less computational cost and found to capture many essential features of 3D convection [30,31], is a useful tool in fundamental research and has been widely used to study Rayleigh-Bénard convection, not only in binary fluids [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27] but also in one-component fluid [28,29,32,33]. Barten and his coworkers [10] first reported the DNS results of convection in water-ethanol mixtures with separation ratio ψ = −0.6, and then studied in detail the TW and SOC states in binary mixtures with different ψ values in a narrow container of a single wavelength [11]. ...
... localized SOC (LSOC, or so-called "convectons") in the 3 He-4 He mixtures for the first time [34], and found that this state also exists in water-ethanol mixtures [15]. After that, multiple LSOC states such as anticonvectons, wall-attached convectons and stable traveling convectons, and collisions between different localized states were studied by different groups [16][17][18][19]. Recently, Smorodin et al. [20] discussed the influence of high-frequency transversal vibrations on the convective instability and pattern formation, and found that the unstable weakly nonlinear TW regime can be stabilized by means of high-frequency vibrations. ...
This paper studied the Rayleigh–Bénard convection in binary fluid mixtures with a strong Soret effect (separation ratio ψ = - 0 . 6 ) in a rectangular container heated uniformly from below. We used a high-accuracy compact finite difference method to solve the hydrodynamic equations used to describe the Rayleigh–Bénard convection. A stable traveling-wave convective state with periodic source defects (PSD-TW) is obtained and its properties are discussed in detail. Our numerical results show that the novel PSD-TW state is maintained by the Eckhaus instability and the difference between the creation and annihilation frequencies of convective rolls at the left and right boundaries of the container. In the range of Rayleigh number in which the PSD-TW state is stable, the period of defect occurrence increases first and then decreases with increasing Rayleigh number. At the upper bound of this range, the system transitions from PSD-TW state to another type of traveling-wave state with aperiodic and more dislocated defects. Moreover, we consider the problem with the Prandtl number P r ranging from 0.1 to 20 and the Lewis number L e from 0.001 to 1, and discuss the stabilities of the PSD-TW states and present the results as phase diagrams.
... 慢的原因 [7,8] , 讨论了Blinking状态以及"重复过 渡"状态(repeated transients)的起源以及长高比 的 影 响 [9] , 也 获 得 了 局 部 定 常 对 流 (localized stationary overturning convection, LSOC) [10] 、 摆 动行波对流(undulation TW, UTW) [11] 和含缺陷 行 波 对 流 [12,13] 等 有 趣 的 流 动 斑 图 . 近 年 来 , Mercader等 [14] 在中等长高比的腔体中, 采用不同 的边界条件观察到了多种LSOC对流结构及其多 重稳定现象, 并发现当上壁面边界改变时奇宇称 LSOC状态整体以一定的速度移动, 且速度大小依 赖于Biot数 [15] . ...
Rayleigh-Bénard (RB) convection in binary fluid mixtures, which shows rich and interesting pattern formation behavior, is a paradigm for understanding instabilities, bifurcations, self-organization with complex spatiotemporal behavior and turbulence, with many applications in atmospheric and environmental physics, astrophysics, and process technology. In this paper, by using a high-order compact finite difference method to solve the full hydrodynamic field equations, we study numerically the RB convection in binary fluid mixtures such as ethanol-water with a very weak Soret effect (separation ratio \begin{document}$\psi=-0.02$\end{document}) in a rectangular container heated uniformly from below. The direct numerical simulations are conducted in the rectangular container with aspect ratio of \begin{document}$\varGamma=12$\end{document} and with four no-slip and impermeable boundaries, isothermal horizontal and perfectly insulated vertical boundaries. The bifurcation and the origin and evolution of pattern in RB convection for the considered physical parameters are studied, and the bifurcation diagram is presented. By performing two-dimensional simulations, we observe three stable states of Blinking state, localized traveling wave and stationary overturning convection (SOC) state, and discuss the transitions between them. The results show that there is a hysteresis in the transition from the Blinking state to the localized traveling wave state for the considered separation ratio, and the evolution of the oscillation frequency, convection amplitude and Nusselt number are discontinuous. Near the lower bound of the Rayleigh number range where the Blinking state exists, a asymmetric initial disturbance is the inducement for the formation of the Blinking state. Inside the range, its inducing effect is weakened, and the oscillatory instability becomes the main reason. It is further confirmed that reflections of lateral walls are responsible for the survival of the stable Blinking state. With the increase of the Rayleigh number, the critical SOC state undergoes multiple bifurcations and forms multiple SOC states with different wave numbers, and then transitions to a chaotic state. There are no stable undulation traveling wave states at both ends of the critical SOC branch.
... We perturb the equation for simplicity by a space-independent noise shaking the system uniformly, which is given by the derivative of a standard real-valued Brownian motion {β(t)} t≥0 . The deterministic (i.e., σ ε = 0), Swift-Hohenberg Equation (2) has been used to model convective systems with mid-plane and left-right reflection symmetries and yields qualitatively reliable predictions of the properties of localized convection in binary fluids [4][5][6][7] and doubly diffusive convection [8]. ...
The purpose of this paper is to rigorously derive the cubic–quintic Ginzburg–Landau equation as a modulation equation for the stochastic Swift–Hohenberg equation with cubic–quintic nonlinearity on an unbounded domain near a change of stability, where a band of dominant pattern is changing stability. Also, we show the influence of degenerate additive noise on the stabilization of the modulation equation.
