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Ecke RE. The center for nonlinear studies: A personal history. CHAOS (WOODBURY, N.Y.) 2024; 34:042102. [PMID: 38639345 DOI: 10.1063/5.0204698] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/22/2024] [Accepted: 03/08/2024] [Indexed: 04/20/2024]
Abstract
The Center for Nonlinear Studies (CNLS) was an integral part of my scientific career starting as a Postdoctoral Fellow in 1983 up to my tenure as CNLS Director from 2004 to 2015. As such, I experienced a number of scientific phases of CNLS through almost four decades of foundation, evolution, and transition. Throughout this entire interval, the inspiration and influence of David Campbell guided my way. A proper history of CNLS encompassing all of the many contributors to the CNLS story is beyond my means or purpose here. Instead, I present the history as I experienced it. I emphasize the main scientific accomplishments achieved at CNLS over more than 40 years, but I will also attempt to describe and quantify the attributes that made and continue to make the Center for Nonlinear Studies a special institution of remarkable impact and longevity. Throughout its existence, CNLS owes much to the enduring legacy of David Campbell who laid down the foundations and operating principles that have made it so successful.
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Affiliation(s)
- Robert E Ecke
- Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
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Ecke RE. Chaos, patterns, coherent structures, and turbulence: Reflections on nonlinear science. CHAOS (WOODBURY, N.Y.) 2015; 25:097605. [PMID: 26428558 DOI: 10.1063/1.4915623] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/05/2023]
Abstract
The paradigms of nonlinear science were succinctly articulated over 25 years ago as deterministic chaos, pattern formation, coherent structures, and adaptation/evolution/learning. For chaos, the main unifying concept was universal routes to chaos in general nonlinear dynamical systems, built upon a framework of bifurcation theory. Pattern formation focused on spatially extended nonlinear systems, taking advantage of symmetry properties to develop highly quantitative amplitude equations of the Ginzburg-Landau type to describe early nonlinear phenomena in the vicinity of critical points. Solitons, mathematically precise localized nonlinear wave states, were generalized to a larger and less precise class of coherent structures such as, for example, concentrated regions of vorticity from laboratory wake flows to the Jovian Great Red Spot. The combination of these three ideas was hoped to provide the tools and concepts for the understanding and characterization of the strongly nonlinear problem of fluid turbulence. Although this early promise has been largely unfulfilled, steady progress has been made using the approaches of nonlinear science. I provide a series of examples of bifurcations and chaos, of one-dimensional and two-dimensional pattern formation, and of turbulence to illustrate both the progress and limitations of the nonlinear science approach. As experimental and computational methods continue to improve, the promise of nonlinear science to elucidate fluid turbulence continues to advance in a steady manner, indicative of the grand challenge nature of strongly nonlinear multi-scale dynamical systems.
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Affiliation(s)
- Robert E Ecke
- Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
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Pal P, Kumar K, Maity P, Dana SK. Pattern dynamics near inverse homoclinic bifurcation in fluids. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:023001. [PMID: 23496604 DOI: 10.1103/physreve.87.023001] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/25/2012] [Indexed: 06/01/2023]
Abstract
We report the pattern dynamics in the vicinity of an inverse homoclinic bifurcation in an extended dissipative system. We observe, in direct numerical simulations of three dimensional Rayleigh-Bénard convection with stress-free top and bottom plates, a spontaneous breaking of a competition of two mutually perpendicular sets of oscillating cross rolls to one of two possible sets of oscillating cross rolls as the Rayleigh number is raised above a critical value. The time period of the oscillating cross-roll patterns diverges and shows scaling behavior near the bifurcation point. This is an example of a transition from nonlocal to local pattern dynamics near an inverse homoclinic bifurcation. We also present a simple four-mode model that captures the pattern dynamics quite well.
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Affiliation(s)
- Pinaki Pal
- Department of Mathematics, National Institute of Technology, Durgapur-713 209, India
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Paul MR, Cross MC, Fischer PF. Rayleigh-Bénard convection with a radial ramp in plate separation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 66:046210. [PMID: 12443302 DOI: 10.1103/physreve.66.046210] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/28/2002] [Indexed: 05/24/2023]
Abstract
Pattern formation in Rayleigh-Bénard convection in a large-aspect-ratio cylinder with a radial ramp in the plate separation is studied analytically and numerically by performing numerical simulations of the Boussinesq equations. A horizontal mean flow and a vertical large scale counterflow are quantified and used to understand the pattern wave number. Our results suggest that the mean flow, generated by amplitude gradients, plays an important role in the roll compression observed as the control parameter is increased. Near threshold, the mean flow has a quadrupole dependence with a single vortex in each quadrant while away from threshold the mean flow exhibits an octupole dependence with a counterrotating pair of vortices in each quadrant. This is confirmed analytically using the amplitude equation and Cross-Newell mean flow equation. By performing numerical experiments, the large scale counterflow is also found to aid in the roll compression away from threshold but to a much lesser degree. Our results yield an understanding of the pattern wave numbers observed in experiment away from threshold and suggest that near threshold the mean flow and large scale counterflow are not responsible for the observed shift to smaller than critical wave numbers.
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Affiliation(s)
- M R Paul
- Department of Physics, California Institute of Technology 114-36, Pasadena 91125, USA.
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Roxin A, Riecke H. Rotating convection in an anisotropic system. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 65:046219. [PMID: 12005988 DOI: 10.1103/physreve.65.046219] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/17/2001] [Indexed: 05/23/2023]
Abstract
We study the stability of patterns arising in rotating convection in weakly anisotropic systems using a modified Swift-Hohenberg equation. The anisotropy, either an endogenous characteristic of the system or induced by external forcing, can stabilize periodic rolls in the Küppers-Lortz chaotic regime. We apply this to the particular case of rotating convection with time-modulated rotation where recently, in experiment, spiral and target patterns have been observed in otherwise Küppers-Lortz-unstable regimes. We show how the underlying base flow breaks the isotropy, thereby affecting the linear growth rate of convection rolls in such a way as to stabilize spirals and targets. Throughout we compare analytical results to numerical simulations of the Swift-Hohenberg equation.
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Affiliation(s)
- Alex Roxin
- Engineering Science and Applied Mathematics, Northwestern University, Evanston, Illinois 60208, USA
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Schmitz R, Pesch W, Zimmermann W. Spiral-defect chaos: Swift-Hohenberg model versus Boussinesq equations. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 65:037302. [PMID: 11909322 DOI: 10.1103/physreve.65.037302] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/19/2001] [Indexed: 05/23/2023]
Abstract
Spiral-defect chaos (SDC) in Rayleigh-Bénard convection is a well-established spatio-temporal complex pattern, which competes with stationary rolls near the onset of convection. The characteristic properties of SDC are accurately described on the basis of the standard three-dimensional Boussinesq equations. As a much simpler and attractive two-dimensional model for SDC generalized Swift-Hohenberg (SH) equations have been extensively used in the literature from the early beginning. Here, we show that the description of SDC by SH models has to be considered with care, especially regarding its long-time dynamics. For parameters used in previous SH simulations, SDC occurs only as a transient in contrast to the experiments and the rigorous solutions of the Boussinesq equations. The small-scale structure of the vorticity field at the spiral cores, which might be crucial for persistent SDC, is presumably not perfectly captured in the SH model.
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Affiliation(s)
- Rainer Schmitz
- Institut für Festkörperforschung, Forschungszentrum Jülich, D-52425 Jülich, Germany
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de Bruyn JR, Lewis BC, Shattuck MD, Swinney HL. Spiral patterns in oscillated granular layers. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 63:041305. [PMID: 11308836 DOI: 10.1103/physreve.63.041305] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/31/2000] [Indexed: 05/23/2023]
Abstract
Cell-filling spiral patterns are observed in a vertically oscillated layer of granular material when the oscillation amplitude is suddenly increased from below the onset of pattern formation into the region where stripe patterns appear for quasistatic increases in amplitude. These spirals are transients and decay to stripe patterns with defects. A transient spiral defect chaos state is also observed. We describe the behavior of the spirals, and the way in which they form and decay. Our results are compared with those for similar spiral patterns in Rayleigh-Bénard convection in fluids.
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Affiliation(s)
- J R de Bruyn
- Department of Physics and Physical Oceanography, Memorial University of Newfoundland, St. John's, Newfoundland, Canada A1B 3X7.
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Matley RG, Wong WY, Thurlow MS, Lucas PG, Lees MJ, Griffiths OJ, Woodcraft AL. Flow pattern dynamics in convecting liquid helium. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2001; 63:045301. [PMID: 11308899 DOI: 10.1103/physreve.63.045301] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/20/2000] [Indexed: 05/23/2023]
Abstract
We present experimental data which correlate thermal measurements and flow visualization in convecting liquid 4He. For a small range R(C)<R<R1 of the Rayleigh number R above the convection threshold value R(C) a robust stationary pattern is observed. This pattern exhibits periodic pulsation when R>R1, generating thermal oscillations as in earlier reports. At higher R values the time dependence becomes aperiodic with the surprising appearance of spiral-defect chaos at an aspect ratio smaller than has previously been reported.
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Affiliation(s)
- R G Matley
- Department of Physics and Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom
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Rudiger S, Feudel F. Pattern formation in rayleigh-Benard convection in a cylindrical container. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 2000; 62:4927-4931. [PMID: 11089038 DOI: 10.1103/physreve.62.4927] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/26/2000] [Indexed: 05/23/2023]
Abstract
We report on numerical investigations of pattern formation in the classical Rayleigh-Benard convection with cylindrical geometry in the regime of low Prandtl numbers and moderate aspect ratio. Beyond the onset of convection, we found straight and bent rolls as stable patterns. By increasing the Rayleigh number, we studied the generation of defects, their dynamics in the form of climbing and gliding, the existence of stable targets and spirals as well as the occurrence of core instabilities, a variety of pattern types that were also observed in experiments.
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Affiliation(s)
- S Rudiger
- Institut fur Physik, Universitat Potsdam, PF 601553, D-14415 Potsdam, Germany
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Dominguez-Lerma MA, Ahlers G, Cannell DS. Rayleigh-Bénard convection in binary mixtures with separation ratios near zero. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1995; 52:6159-6174. [PMID: 9964133 DOI: 10.1103/physreve.52.6159] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Xi H, Gunton JD. Spatiotemporal chaos in a model of Rayleigh-Bénard convection. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1995; 52:4963-4975. [PMID: 9963992 DOI: 10.1103/physreve.52.4963] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Hu Y, Ecke R, Ahlers G. Convection for Prandtl numbers near 1: Dynamics of textured patterns. PHYSICAL REVIEW. E, STATISTICAL PHYSICS, PLASMAS, FLUIDS, AND RELATED INTERDISCIPLINARY TOPICS 1995; 51:3263-3279. [PMID: 9963004 DOI: 10.1103/physreve.51.3263] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Hu Y, Ecke RE, Ahlers G. Transition to spiral-defect chaos in low Prandtl number convection. PHYSICAL REVIEW LETTERS 1995; 74:391-394. [PMID: 10058746 DOI: 10.1103/physrevlett.74.391] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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