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Persistent patterns and multifractality in fluid mixing Bala Sundaram,1 Andrew C. Poje,2 and Arjendu K. Pattanayak3 1Department of Physics, University of Massachusetts, 100 Morrissey Boulevard, Boston, Massachusetts 02125, USA 2Department of Mathematics and Graduate Faculty in Physics, City University of New YorkCSI, Staten Island, New York 10314, USA 3Department of Physics and Astronomy, Carleton College, Northfield, Minnesota 55057, USA Received 23 December 2008; revised manuscript received 12 May 2009; published 5 June 2009 Persistent patterns in periodically driven dynamics have been reported in a wide variety of contexts ranging from tabletop and oceanscale fluid mixing systems to the weak quantumclassical transition in open Hamiltonian systems. We illustrate a common framework for the emergence of these patterns by considering a simple measure of structure maintenance provided by the average radius of the scalar distribution in transform space. Within this framework, scaling laws related to both the formation and persistence of patterns in phase space are presented. Further, preliminary results linking the scaling exponents associated with the persistent patterns to the multifractal nature of the advective phasespace geometry are shown. DOI: 10.1103/PhysRevE.79.066202 PACS number s : 05.45. a, 47.52. j, 47.54. r, 83.50.Xa I. INTRODUCTION The role of chaotic dynamics in assisting the mixing of a passive scalar is of broad interest with applications ranging from micromixers to chemical reactors to geophysical tracer transport. These dynamics are a stirring mechanism, which conventional wisdom suggests facilitates the mixing process 1 . The evolution of a passive scalar field is described by the advectiondiffusion equation C t + u · C = D 2C, 1 where C x , t is the concentration, D is the constant diffusivity, and u is the prescribed Eulerian velocity. The two processes of advective mixing and diffusion work very differently: stretching and folding by the chaotic velocity field rapidly sharpens concentration gradients, while diffusion acts to smooth these gradients. As a result, the competition between the two can lead to counterintuitive results. In particular, a number of experimental and numerical investigations 2–7 has shown that the longtime dynamics of the scalar field under the action of timeperiodic chaotic advection is itself time periodic after appropriate rescaling of the variance and is completely determined by the slowest decaying Floquet mode of the corresponding single period advectiondiffusion Poincaré operator. This “strange eigenmode” 2 is characterized by exponential decay of the scalar variance and selfsimilar evolution of both the scalar spectrum and probability distribution. Studies of these patterns show that regions of high concentration gradients of the passive scalar are associated with features of the underlying chaos in the Lagrangian dynamics, specifically the unstable manifolds of the fixed points of associated Poincaré maps 4 and the boundaries of integrable regions of the flow produced by the presence of KolmogorovArnoldMoser tori 7 . Interest in the nature of such persistent patterns, extends beyond fluid dynamics and mixing. Specifically, Eq. 1 in even dimensions corresponds to the dynamics of a classical probability density in the presence of noise. The dynamics of the relaxation of such densities to equilibrium is of relevance to the foundations of statistical mechanics. Further, the rate of decay of gradients in classical probabilities strongly influences the difference between quantum and classical evolutions, thus making these patterns of interest in quantumclassical correspondence 8 as well. Specific questions concerning what sets the overall decay rate of the persistent pattern and how this decay rate scales with both the diffusivity and parameters in the velocity field have been addressed in a number of flows, typically those representable by discrete twodimensional 2D maps. The range of validity of local theories based on the distribution of stretching rates in the chaotic field 9,10 has been investigated for homogeneous or nearly homogeneous maps 5,11–13 . Direct calculation of the spectral properties of the Poincaré operator of the advectiondiffusion equation for maps with mixed phasespace dynamics has shown that the degree of spatial localization of eigensolutions, the scaling of the spectrum with diffusivity and the existence of degenerate states are intimately connected with finescale details of the underlying phasespace geometry 7,14 . In this paper we examine the connection between measurable statistical properties of the geometry of the advective dynamics and the scaling of a previously proposed, easily computed, measure of pattern strength 15 . This measure, defined by the Dirichlet quotient of the scalar energy, L2 t = C x , t 2dx , and enstrophy, C2 t = C x , t 2dx , 2 t = C2 t L2 t = k 2 C k ,t 2dk C k ,t 2dk , 2 explicitly tracks the competition between advective and diffusive effects. Physically, 2 t measures the meansquare radius of the scalar distribution in k space, reflecting the distance between the largest stirring scales and the smallest length scales at which advection can produce structure in the presence of smoothing. Previous analysis 15 concentrated on uniformly hyperbolic chaotic systems, where it was shown that 2 t initially grows as the dynamics produces structure, reaches a plateau which scales as D−1 due to a balance between the dynamics and diffusion, and then PHYSICAL REVIEW E 79, 066202 2009 15393755/2009/79 6 /066202 7 0662021 ©2009 The American Physical Society
Object Description
Collection Title  Scholarly Publications by Carleton Faculty and Staff 
Journal Title  Physical Review E 
Article Title  Persistent patterns and multifractality in fluid mixing 
Article Author 
Pattanayak, Arjendu Sundaram, Bala Poje, Andrew 
Carleton Author 
Pattanayak, Arjendu 
Department  Physics 
Field  Science and Mathematics 
Year  2009 
Volume  79 
Publisher  American Physical Society 
File Name  049_PattanayakArjendu_PersistentPatternsAndMultifractalityInFluidMixing.pdf; 049_PattanayakArjendu_PersistentPatternsAndMultifractalityInFluidMixing.pdf 
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Contributing Organization  Carleton College 
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Language  English 
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Article Title  Page 1 
FullText  Persistent patterns and multifractality in fluid mixing Bala Sundaram,1 Andrew C. Poje,2 and Arjendu K. Pattanayak3 1Department of Physics, University of Massachusetts, 100 Morrissey Boulevard, Boston, Massachusetts 02125, USA 2Department of Mathematics and Graduate Faculty in Physics, City University of New YorkCSI, Staten Island, New York 10314, USA 3Department of Physics and Astronomy, Carleton College, Northfield, Minnesota 55057, USA Received 23 December 2008; revised manuscript received 12 May 2009; published 5 June 2009 Persistent patterns in periodically driven dynamics have been reported in a wide variety of contexts ranging from tabletop and oceanscale fluid mixing systems to the weak quantumclassical transition in open Hamiltonian systems. We illustrate a common framework for the emergence of these patterns by considering a simple measure of structure maintenance provided by the average radius of the scalar distribution in transform space. Within this framework, scaling laws related to both the formation and persistence of patterns in phase space are presented. Further, preliminary results linking the scaling exponents associated with the persistent patterns to the multifractal nature of the advective phasespace geometry are shown. DOI: 10.1103/PhysRevE.79.066202 PACS number s : 05.45. a, 47.52. j, 47.54. r, 83.50.Xa I. INTRODUCTION The role of chaotic dynamics in assisting the mixing of a passive scalar is of broad interest with applications ranging from micromixers to chemical reactors to geophysical tracer transport. These dynamics are a stirring mechanism, which conventional wisdom suggests facilitates the mixing process 1 . The evolution of a passive scalar field is described by the advectiondiffusion equation C t + u · C = D 2C, 1 where C x , t is the concentration, D is the constant diffusivity, and u is the prescribed Eulerian velocity. The two processes of advective mixing and diffusion work very differently: stretching and folding by the chaotic velocity field rapidly sharpens concentration gradients, while diffusion acts to smooth these gradients. As a result, the competition between the two can lead to counterintuitive results. In particular, a number of experimental and numerical investigations 2–7 has shown that the longtime dynamics of the scalar field under the action of timeperiodic chaotic advection is itself time periodic after appropriate rescaling of the variance and is completely determined by the slowest decaying Floquet mode of the corresponding single period advectiondiffusion Poincaré operator. This “strange eigenmode” 2 is characterized by exponential decay of the scalar variance and selfsimilar evolution of both the scalar spectrum and probability distribution. Studies of these patterns show that regions of high concentration gradients of the passive scalar are associated with features of the underlying chaos in the Lagrangian dynamics, specifically the unstable manifolds of the fixed points of associated Poincaré maps 4 and the boundaries of integrable regions of the flow produced by the presence of KolmogorovArnoldMoser tori 7 . Interest in the nature of such persistent patterns, extends beyond fluid dynamics and mixing. Specifically, Eq. 1 in even dimensions corresponds to the dynamics of a classical probability density in the presence of noise. The dynamics of the relaxation of such densities to equilibrium is of relevance to the foundations of statistical mechanics. Further, the rate of decay of gradients in classical probabilities strongly influences the difference between quantum and classical evolutions, thus making these patterns of interest in quantumclassical correspondence 8 as well. Specific questions concerning what sets the overall decay rate of the persistent pattern and how this decay rate scales with both the diffusivity and parameters in the velocity field have been addressed in a number of flows, typically those representable by discrete twodimensional 2D maps. The range of validity of local theories based on the distribution of stretching rates in the chaotic field 9,10 has been investigated for homogeneous or nearly homogeneous maps 5,11–13 . Direct calculation of the spectral properties of the Poincaré operator of the advectiondiffusion equation for maps with mixed phasespace dynamics has shown that the degree of spatial localization of eigensolutions, the scaling of the spectrum with diffusivity and the existence of degenerate states are intimately connected with finescale details of the underlying phasespace geometry 7,14 . In this paper we examine the connection between measurable statistical properties of the geometry of the advective dynamics and the scaling of a previously proposed, easily computed, measure of pattern strength 15 . This measure, defined by the Dirichlet quotient of the scalar energy, L2 t = C x , t 2dx , and enstrophy, C2 t = C x , t 2dx , 2 t = C2 t L2 t = k 2 C k ,t 2dk C k ,t 2dk , 2 explicitly tracks the competition between advective and diffusive effects. Physically, 2 t measures the meansquare radius of the scalar distribution in k space, reflecting the distance between the largest stirring scales and the smallest length scales at which advection can produce structure in the presence of smoothing. Previous analysis 15 concentrated on uniformly hyperbolic chaotic systems, where it was shown that 2 t initially grows as the dynamics produces structure, reaches a plateau which scales as D−1 due to a balance between the dynamics and diffusion, and then PHYSICAL REVIEW E 79, 066202 2009 15393755/2009/79 6 /066202 7 0662021 ©2009 The American Physical Society 