1
|
Faranda D, Leoncini X, Vaienti S. Mixing properties in the advection of passive tracers via recurrences and extreme value theory. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 89:052901. [PMID: 25353855 DOI: 10.1103/physreve.89.052901] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/20/2014] [Indexed: 06/04/2023]
Abstract
In this paper we characterize the mixing properties in the advection of passive tracers by exploiting the extreme value theory for dynamical systems. With respect to classical techniques directly related to the Poincaré recurrences analysis, our method provides reliable estimations of the characteristic mixing times and distinguishes between barriers and unstable fixed points. The method is based on a check of convergence for extreme value laws on finite datasets. We define the mixing times in terms of the shortest time intervals such that extremes converge to the asymptotic (known) parameters of the generalized extreme value distribution. Our technique is suitable for applications in the analysis of other systems where mixing time scales need to be determined and limited datasets are available.
Collapse
Affiliation(s)
- Davide Faranda
- Laboratoire SPHYNX, Service de Physique de l'Etat Condensé, DSM, CEA Saclay, CNRS URA 2464, 91191 Gif-sur-Yvette, France
| | - Xavier Leoncini
- Aix Marseille Université, CNRS, CPT, UMR 7332, 13288 Marseille, France and Université de Toulon, CNRS, CPT, UMR 7332, 83957 La Garde, France
| | - Sandro Vaienti
- Aix Marseille Université, CNRS, CPT, UMR 7332, 13288 Marseille, France and Université de Toulon, CNRS, CPT, UMR 7332, 83957 La Garde, France
| |
Collapse
|
2
|
Bargteil D, Solomon T. Barriers to front propagation in ordered and disordered vortex flows. CHAOS (WOODBURY, N.Y.) 2012; 22:037103. [PMID: 23020494 DOI: 10.1063/1.4746764] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/01/2023]
Abstract
We present experiments on reactive front propagation in a two-dimensional (2D) vortex chain flow (both time-independent and time-periodic) and a 2D spatially disordered (time-independent) vortex-dominated flow. The flows are generated using magnetohydrodynamic forcing techniques, and the fronts are produced using the excitable, ferroin-catalyzed Belousov-Zhabotinsky chemical reaction. In both of these flows, front propagation is dominated by the presence of burning invariant manifolds (BIMs) that act as barriers, similar to invariant manifolds that dominate the transport of passive impurities. Convergence of the fronts onto these BIMs is shown experimentally for all of the flows studied. The BIMs are also shown to collapse onto the invariant manifolds for passive transport in the limit of large flow velocities. For the disordered flow, the measured BIMs are compared to those predicted using a measured velocity field and a three-dimensional set of ordinary differential equations that describe the dynamics of front propagation in advection-reaction-diffusion systems.
Collapse
Affiliation(s)
- Dylan Bargteil
- Department of Physics and Astronomy, Bucknell University, Lewisburg, Pennsylvania 17837, USA
| | | |
Collapse
|
3
|
Takeuchi KA, Kuroda M, Chaté H, Sano M. Experimental realization of directed percolation criticality in turbulent liquid crystals. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 80:051116. [PMID: 20364956 DOI: 10.1103/physreve.80.051116] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/24/2009] [Indexed: 05/29/2023]
Abstract
This is a comprehensive report on the phase transition between two turbulent states of electroconvection in nematic liquid crystals, which was recently found by the authors to be in the directed percolation (DP) universality class [K. A. Takeuchi, Phys. Rev. Lett. 99, 234503 (2007)]. We further investigate both static and dynamic critical behaviors of this phase transition, measuring a total of 12 critical exponents, 5 scaling functions, and 8 scaling relations, all in full agreement with those characterizing the DP class in 2+1 dimensions. Developing an experimental technique to create a seed of topological-defect turbulence by pulse laser, we confirm in particular the rapidity symmetry, which is a basic but nontrivial consequence of the field-theoretic approach to DP. This provides a clear experimental realization of this outstanding truly out-of-equilibrium universality class, dominating most phase transitions into an absorbing state.
Collapse
Affiliation(s)
- Kazumasa A Takeuchi
- Department of Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan.
| | | | | | | |
Collapse
|
4
|
Takeuchi KA, Kuroda M, Chaté H, Sano M. Directed percolation criticality in turbulent liquid crystals. PHYSICAL REVIEW LETTERS 2007; 99:234503. [PMID: 18233372 DOI: 10.1103/physrevlett.99.234503] [Citation(s) in RCA: 112] [Impact Index Per Article: 6.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/28/2007] [Indexed: 05/25/2023]
Abstract
We experimentally investigate the critical behavior of a phase transition between two topologically different turbulent states of electrohydrodynamic convection in nematic liquid crystals. The statistical properties of the observed spatiotemporal intermittency regimes are carefully determined, yielding a complete set of static critical exponents in full agreement with those defining the directed percolation class in 2+1 dimensions. This constitutes the first clear and comprehensive experimental evidence of an absorbing phase transition in this prominent nonequilibrium universality class.
Collapse
Affiliation(s)
- Kazumasa A Takeuchi
- Department of Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan.
| | | | | | | |
Collapse
|
5
|
Bachelard R, Benzekri T, Chandre C, Leoncini X, Vittot M. Targeted mixing in an array of alternating vortices. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2007; 76:046217. [PMID: 17995092 DOI: 10.1103/physreve.76.046217] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/16/2007] [Indexed: 05/25/2023]
Abstract
Transport and mixing properties of passive particles advected by an array of vortices are investigated. Starting from the integrable case, it is shown that a special class of perturbations allows one to preserve separatrices which act as effective transport barriers, while triggering chaotic advection. In this setting, mixing within the two dynamical barriers is enhanced while long range transport is prevented. A numerical analysis of mixing properties depending on parameter values is performed; regions for which optimal mixing is achieved are proposed. Robustness of the targeted mixing properties regarding errors in the applied perturbation are considered, as well as slip/no-slip and/or boundary conditions for the flow.
Collapse
Affiliation(s)
- R Bachelard
- Centre de Physique Théorique, CNRS Aix-Marseille Universités, campus de Luminy, Case 907, F-13288 Marseille cedex 9, France
| | | | | | | | | |
Collapse
|
6
|
Pocheau A, Harambat F. Effective front propagation in steady cellular flows: A least time criterion. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 73:065304. [PMID: 16906901 DOI: 10.1103/physreve.73.065304] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/07/2005] [Indexed: 05/11/2023]
Abstract
We experimentally study the propagation of a reaction front in a chain of counter-rotating vortices. The front is induced by an autocatalytic chemical reaction in an aqueous solution stirred by electroconvective means. The front propagates by getting quickly engulfed in a vortex and by slowly crossing the separatrix to the next vortex. Its mean velocity along the chain increases with the flow intensity but shows both a bending and a dependence on the vortex aspect ratio. We recover these features within a kinematic model of front propagation by seeking the quickest front path through a vortex.
Collapse
Affiliation(s)
- A Pocheau
- IRPHE, CNRS et Universités Aix-Marseille I & II, 49 rue Joliot-Curie, B.P. 146, Technopôle de Château-Gombert, F-13384 Marseille, Cedex 13, France
| | | |
Collapse
|
7
|
Benzekri T, Chandre C, Leoncini X, Lima R, Vittot M. Chaotic advection and targeted mixing. PHYSICAL REVIEW LETTERS 2006; 96:124503. [PMID: 16605911 DOI: 10.1103/physrevlett.96.124503] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/06/2005] [Indexed: 05/08/2023]
Abstract
The advection of passive tracers in an oscillating vortex chain is investigated. It is shown that by adding a suitable perturbation to the ideal flow, the induced chaotic advection exhibits two remarkable properties which do not hold in the case of a generic perturbation: Particles remain trapped within a specific domain bounded by two oscillating barriers (suppression of chaotic transport along the channel), and the stochastic sea seems to cover this whole bounded domain (enhancement of mixing within the rolls).
Collapse
Affiliation(s)
- T Benzekri
- Centre de Physique Théorique - UMR 6207, CNRS Luminy, case 907, F-13288 Marseille cedex 09, France
| | | | | | | | | |
Collapse
|
8
|
Paoletti MS, Nugent CR, Solomon TH. Synchronization of oscillating reactions in an extended fluid system. PHYSICAL REVIEW LETTERS 2006; 96:124101. [PMID: 16605908 DOI: 10.1103/physrevlett.96.124101] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/30/2005] [Indexed: 05/08/2023]
Abstract
We present experiments on the synchronization of a dynamical, chemical process in an extended, flowing, fluid system. The oscillatory Belousov-Zhabotinsky chemical reaction is the process studied, and the flow is an annular chain of counterrotating vortices. Azimuthal motion of the vortices is controlled externally, enabling us to vary the type of transport. We find that oscillations of the Belousov-Zhabotinsky reaction synchronize throughout the extended fluid system only if transport in the flow is superdiffusive, with tracers in the flow undergoing rapid, distant jumps called Lévy flights.
Collapse
Affiliation(s)
- M S Paoletti
- Department of Physics, Bucknell University, Lewisburg, Pennsylvania 17837, USA
| | | | | |
Collapse
|
9
|
Paoletti MS, Solomon TH. Front propagation and mode-locking in an advection-reaction-diffusion system. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 72:046204. [PMID: 16383509 DOI: 10.1103/physreve.72.046204] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/22/2005] [Indexed: 05/05/2023]
Abstract
Experiments are presented on chemical front propagation in an oscillating chain of vortices in which the mixing of passive impurities is chaotic. The excitable ruthenium-catalyzed Belousov-Zhabotinsky reaction is used in these studies. Velocities of the propagating fronts are measured as a function of the frequency and amplitude of external forcing. Mode locking is observed where the front propagates an integer number of vortices in an integer number of drive periods. Arnol'd tongues are mapped out for two of the locking regimes. These two tongues are shown to form a region of overlap where the velocity of the propagating front switches erratically between two locked values. The experimental results agree with numerical predictions of mode locking in a simplified model of the flow.
Collapse
Affiliation(s)
- M S Paoletti
- Department of Physics, Bucknell University, Lewisburg, Pennsylvania 17837, USA
| | | |
Collapse
|
10
|
Mohamadou A, Jiotsa AK, Kofané TC. Modulational instability and unstable patterns in the discrete complex cubic Ginzburg-Landau equation with first and second neighbor couplings. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2005; 72:036220. [PMID: 16241561 DOI: 10.1103/physreve.72.036220] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/03/2003] [Revised: 04/18/2005] [Indexed: 05/05/2023]
Abstract
The generation of nonlinear modulated waves is investigated in the framework of hydrodynamics using a model of coupled oscillators. In this model, the separatrices between each pair of vortices may be viewed as individual oscillators and are described by a phenomenological one-dimensional discrete complex Ginzburg-Landau equation involving first- and second-nearest neighbor couplings. A theoretical approach based on the linear stability analysis predicts regions of modulational instability, governed by both the first and second-nearest neighbor couplings. From numerical investigations of different wave patterns that may be driven by the modulational instability, it appears that analytical predictions are correctly verified. For wave number in the unstable regions, an initial condition whose amplitude is slightly modulated breaks into a train of unstable patterns. This phenomenon agrees with the description of amplification of the spectral component of the perturbation and its harmonics, as well.
Collapse
Affiliation(s)
- Alidou Mohamadou
- Laboratoire de Mécanique, Département de Physique, Faculté des Sciences, Université de Yaoundé I, B.P. 812 Yaoundé, Cameroun
| | | | | |
Collapse
|
11
|
Nugent CR, Quarles WM, Solomon TH. Experimental studies of pattern formation in a reaction-advection-diffusion system. PHYSICAL REVIEW LETTERS 2004; 93:218301. [PMID: 15601066 DOI: 10.1103/physrevlett.93.218301] [Citation(s) in RCA: 13] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/20/2004] [Indexed: 05/24/2023]
Abstract
Experiments are presented on pattern formation in the Belousov-Zhabotinsky (BZ) reaction in a blinking vortex flow. Mixing in this flow is chaotic, with nearby tracers separating exponentially with time. The patterns that form in this flow with the BZ reaction mimic chaotic mixing structures seen in passive transport. The behavior is analyzed in terms of a mixing time taum and a characteristic decorrelation time TBZ for the BZ system. Flows with taum comparable to or smaller than TBZ generate large-scale patterns whose features are captured by simulations of mixing fields for the flow.
Collapse
Affiliation(s)
- C R Nugent
- Department of Physics, Bucknell University, Lewisburg, Pennsylvania 17837, USA
| | | | | |
Collapse
|
12
|
Trueba JL, Baltanás JP, Feudel F, Sanjuán MAF. On the estimate of the stochastic layer width for a model of tracer dynamics. CHAOS (WOODBURY, N.Y.) 2003; 13:866-873. [PMID: 12946178 DOI: 10.1063/1.1598151] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
An analytical estimate of the width of the generated chaotic layer in a time-periodically driven stream function model for the motion of passive tracers is discussed. It is based essentially on the method of the separatrix map and the use of the Melnikov theory. Energy-time variables are used to derive lower bounds for the half width of the layer. In order to perform a comparison with numerical simulations, the results are transformed into space variables. The analytic results of the layer thickness in both parallel and perpendicular directions to the shear flow are compared with numerical computations and some systematic deviations are discussed.
Collapse
Affiliation(s)
- José L Trueba
- Nonlinear Dynamics and Chaos Group, Departamento de Matemáticas y Física Aplicadas y Ciencias de la Naturaleza, Universidad Rey Juan Carlos, Tulipán s/n, 28933 Móstoles, Madrid, Spain
| | | | | | | |
Collapse
|
13
|
Rupp P, Richter R, Rehberg I. Critical exponents of directed percolation measured in spatiotemporal intermittency. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 67:036209. [PMID: 12689155 DOI: 10.1103/physreve.67.036209] [Citation(s) in RCA: 14] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/11/2002] [Revised: 10/31/2002] [Indexed: 05/24/2023]
Abstract
An experimental system showing a transition to spatiotemporal intermittency is presented. It consists of a ring of hundred oscillating ferrofluidic spikes. Four of five of the measured critical exponents of the system agree with those obtained from a theoretical model of directed percolation.
Collapse
Affiliation(s)
- Peter Rupp
- Physikalisches Institut, Experimentalphysik V, Universität Bayreuth, Germany
| | | | | |
Collapse
|
14
|
Fogleman MA, Fawcett MJ, Solomon TH. Lagrangian chaos and correlated Lévy flights in a non-Beltrami flow: transient versus long-term transport. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 63:020101. [PMID: 11308448 DOI: 10.1103/physreve.63.020101] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/19/2000] [Indexed: 05/23/2023]
Abstract
Long-range transport is studied numerically in a time-independent, three-dimensional (3D) fluid flow composed of the superposition of two chains of alternating vortices, one horizontal and the other vertical. Tracers in this flow follow chaotic trajectories composed of correlated Lévy flights with varying velocities. Locations of the chaotic regimes in the flow are compared with recent theories of chaos in non-Beltrami 3D flows. Growth of the variance of a distribution of tracers is divided into transient and long-term regimes, each with different growth exponents.
Collapse
Affiliation(s)
- M A Fogleman
- Department of Physics, Bucknell University, Lewisburg, Pennsylvania 17837, USA
| | | | | |
Collapse
|
15
|
Solomon TH, Hartley RR, Lee AT. Aggregation and chimney formation during the solidification of ammonium chloride. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1999; 60:3063-71. [PMID: 11970113 DOI: 10.1103/physreve.60.3063] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/06/1998] [Revised: 05/18/1999] [Indexed: 11/07/2022]
Abstract
Experiments study large-scale pattern formation during the growth of ammonium chloride (NH4Cl) from solution in a thin (Hele-Shaw) geometry. In particular a solid-liquid mixture ("mushy layer") forms in which growing solid NH4Cl crystals form a solid network interspersed with liquid. There are different ways that the mushy layer can be formed, however. If the cell is heated from below and cooled from above, thermal convection generates large-scale recirculating flows that carry seed crystals from the upper (cold) boundary to the (warmer) side and bottom boundaries. Ballistic deposition of these seed crystals leads to aggregation patterns with significant voids (filled with liquid) with a wide range of length scales. If the cell is cooled from below with a warm environment, the solid NH4Cl grows dendritically without deposition, resulting in a compact mushy layer. Plume convection within this mushy layer produces one or two well-defined "chimneys." If the environment is cool (comparable to the liquidus temperature of the solution), the mushy layer forms by a combination of dendritic growth and ballistic deposition, resulting in a more permeable mushy layer and enhanced chimney formation. The effects of ballistic deposition are enhanced if the cell is tipped, in which case the voids reappear. Plume convection and chimney formation are dramatically enhanced in this case. Additional experiments are done in which fluid flows in the system are enhanced artificially to verify that enhancements in chimney formation are due primarily to the aggregation process, and not to the increases in fluid flows due to thermal and compositional convection.
Collapse
Affiliation(s)
- T H Solomon
- Department of Physics, Bucknell University, Lewisburg, Pennsylvania 17837, USA.
| | | | | |
Collapse
|
16
|
Solomon TH, Tomas S, Warner JL. Role of Lobes in Chaotic Mixing of Miscible and Immiscible Impurities. PHYSICAL REVIEW LETTERS 1996; 77:2682-2685. [PMID: 10062019 DOI: 10.1103/physrevlett.77.2682] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
|
17
|
Degen MM, Mutabazi I. Transition to weak turbulence via spatiotemporal intermittency in the Taylor-Dean system. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1996; 53:3495-3504. [PMID: 9964659 DOI: 10.1103/physreve.53.3495] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
|