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Abstract
A study on an advectiondiffusionreaction system is presented. Variability of the reaction process in such a system triggered by a highly localized source is quantified. It is found, for geophysically motivated parameter regimes, that the difference in bulk concentration subject to realizations of different source locations is highly correlated with the local flow topology of the source. Such flow topologies can be highlighted by Lagrangian coherent structures. Reaction is relatively enhanced in regions of strong stretching, and relatively suppressed in regions where vortices are present. In any case, the presence of a divergencefree background flow helps speed up the reaction process, especially when the flow is timedependent. Probability density of various quantities characterizing the reaction processes is also obtained. This reveals the inherent complexity of the reactiondiffusion process subject to nonlinear background stirring.
We acknowledge the support from Grant No. NSFDMS1212144. We also thank two anonymous referees for their valuable comments, and the many helpful discussions with Emilio HernándezGarcía and Tom Solomon.
I. INTRODUCTION
II. BACKGROUND
A. Lagrangian coherent structures and the kinematic flow model
B. The governing equation and methodology
III. RESULTS
A. Variability on reaction
B. Reaction and Lagrangian stretching
C. Parametric dependence on LCS
D. Dependence of variability on governing parameters
IV. DISCUSSIONS AND CONCLUSIONS
Key Topics
 Liquid crystals
 37.0
 Diffusion
 20.0
 Lagrangian mechanics
 18.0
 Chemically reactive flows
 17.0
 Manifolds
 16.0
Figures
Range of behaviors of the reaction process. (a) Variation of the average concentration ⟨c⟩ in the periodic domain. The two black curves are the bounding cases of the fastest and slowest reaction behaviors among the realizations. (b) Probability density of ⟨c⟩ at T = 7.5. The range of behavior lies on the vertical dashed line in (a). (c) Probability density of the reaction time when ⟨c⟩ = 0.5. The range of behavior lies on the horizontal solid line in (a).
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Range of behaviors of the reaction process. (a) Variation of the average concentration ⟨c⟩ in the periodic domain. The two black curves are the bounding cases of the fastest and slowest reaction behaviors among the realizations. (b) Probability density of ⟨c⟩ at T = 7.5. The range of behavior lies on the vertical dashed line in (a). (c) Probability density of the reaction time when ⟨c⟩ = 0.5. The range of behavior lies on the horizontal solid line in (a).
Comparison between the variability in ⟨c⟩ and LCS. Each point in (a) marks the center of a concentration blob initially released in the flow (a realization). The color value is the associated ⟨c⟩ of this realization at T = 7.5. (b) Repelling LCS marked by the forwardtime FTLE field. The four colored dots mark the initial conditions of four representative realizations discussed later. Note the range of ⟨c⟩ at this time and the strong correlation between the two panels.
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Comparison between the variability in ⟨c⟩ and LCS. Each point in (a) marks the center of a concentration blob initially released in the flow (a realization). The color value is the associated ⟨c⟩ of this realization at T = 7.5. (b) Repelling LCS marked by the forwardtime FTLE field. The four colored dots mark the initial conditions of four representative realizations discussed later. Note the range of ⟨c⟩ at this time and the strong correlation between the two panels.
Range of behaviors of the reaction process. (a) Placement of the reaction processes in the chaotic flow with other flow types. The inset plot shows the earlytime behavior of these cases. (b) Difference in reaction rates between the actual realizations and the nominal logistic equation case red line). The inset plot shows the early time behavior in log scale. The vertical line is a reference line at ⟨c⟩ = 0.0035. This line is also shown in the inset plot in (a). The blue line is the realization with no flow. (c) Correlation between the structures in Fig. 2 as a function of time. The inset plot shows the comparison between probability density of FTLE (black) and ⟨c⟩ (blue). The black horizontal bar marks the range of FTLE between 2 and 4. The blue horizontal bar marks the range of ⟨c⟩ between 0.52 and 0.55.
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Range of behaviors of the reaction process. (a) Placement of the reaction processes in the chaotic flow with other flow types. The inset plot shows the earlytime behavior of these cases. (b) Difference in reaction rates between the actual realizations and the nominal logistic equation case red line). The inset plot shows the early time behavior in log scale. The vertical line is a reference line at ⟨c⟩ = 0.0035. This line is also shown in the inset plot in (a). The blue line is the realization with no flow. (c) Correlation between the structures in Fig. 2 as a function of time. The inset plot shows the comparison between probability density of FTLE (black) and ⟨c⟩ (blue). The black horizontal bar marks the range of FTLE between 2 and 4. The blue horizontal bar marks the range of ⟨c⟩ between 0.52 and 0.55.
Chemical concentration at T = 1. The four realizations are initially released at (a) black dot, (b) white dot, (c) red dot, and (d) magenta dot in Fig. 2 . The black (white) curves are the isocontours for forward (backward)time FTLE indicating repelling (attracting) structures. The titles indicate the average chemical content ⟨c⟩.
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Chemical concentration at T = 1. The four realizations are initially released at (a) black dot, (b) white dot, (c) red dot, and (d) magenta dot in Fig. 2 . The black (white) curves are the isocontours for forward (backward)time FTLE indicating repelling (attracting) structures. The titles indicate the average chemical content ⟨c⟩.
Same as Fig. 4 but at T = 7.5. Note the major difference between (b) and the other cases (a, c, and d).
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Same as Fig. 4 but at T = 7.5. Note the major difference between (b) and the other cases (a, c, and d).
Evolution of individual contributions from Eqs. (6) and (7) to the scalar variance ⟨c⟩^{2}, , and ⟨c ^{2}⟩, as functions of ⟨c⟩. (a) Terms relevant to Eq. (6) . The black curve is ⟨c⟩^{2}, which is a straight line in log scale as a function of ⟨c⟩. The blue curve is 2η⟨c⟩^{2} − 2η⟨c⟩^{3}. The green curves are for the four cases, plotted in thick solid (hyperbolic core), thick dashed (interior), dasheddotted (repeller), and dotted (attractor) lines. The red vertical line references ⟨c⟩ = 0.002, where all cases are around T = 1. (b) Terms relevant to Eq. (7) . The black curves are . The blue curves are and the green curves are 2⟨ ∇ c p ^{2}⟩/Pe. Line styles are the same as those in panel (a). The red vertical line again marks ⟨c⟩ = 0.002. (c) Terms relevant to evolution of ⟨c ^{2}⟩. The black curves are ⟨c ^{2}⟩, the blue curves are 2η⟨c ^{2}⟩ − 2η⟨c ^{3}⟩ and the green curves are 2⟨ ∇ c p ^{2}⟩/Pe. Line styles are the same as those in panel (a). The red solid vertical line marks ⟨c⟩ = 0.002, the three red dashed vertical lines mark ⟨c⟩ = 0.001, 0.01, and 0.1, respectively. All realizations reach ⟨c⟩ = 0.1 between T = 5 and 5.5.
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Evolution of individual contributions from Eqs. (6) and (7) to the scalar variance ⟨c⟩^{2}, , and ⟨c ^{2}⟩, as functions of ⟨c⟩. (a) Terms relevant to Eq. (6) . The black curve is ⟨c⟩^{2}, which is a straight line in log scale as a function of ⟨c⟩. The blue curve is 2η⟨c⟩^{2} − 2η⟨c⟩^{3}. The green curves are for the four cases, plotted in thick solid (hyperbolic core), thick dashed (interior), dasheddotted (repeller), and dotted (attractor) lines. The red vertical line references ⟨c⟩ = 0.002, where all cases are around T = 1. (b) Terms relevant to Eq. (7) . The black curves are . The blue curves are and the green curves are 2⟨ ∇ c p ^{2}⟩/Pe. Line styles are the same as those in panel (a). The red vertical line again marks ⟨c⟩ = 0.002. (c) Terms relevant to evolution of ⟨c ^{2}⟩. The black curves are ⟨c ^{2}⟩, the blue curves are 2η⟨c ^{2}⟩ − 2η⟨c ^{3}⟩ and the green curves are 2⟨ ∇ c p ^{2}⟩/Pe. Line styles are the same as those in panel (a). The red solid vertical line marks ⟨c⟩ = 0.002, the three red dashed vertical lines mark ⟨c⟩ = 0.001, 0.01, and 0.1, respectively. All realizations reach ⟨c⟩ = 0.1 between T = 5 and 5.5.
Phases of the entire process for two realizations. Top row: release at hyperbolic core. Bottom row: release at interior. Left column: end of the first phase dominant by scalar decay. Center column: end of the second phase dominant by filament stretching. Right column: end of the third phase dominant by homogenization among neighboring filaments. End of the fourth phase is uniform concentration and is not shown.
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Phases of the entire process for two realizations. Top row: release at hyperbolic core. Bottom row: release at interior. Left column: end of the first phase dominant by scalar decay. Center column: end of the second phase dominant by filament stretching. Right column: end of the third phase dominant by homogenization among neighboring filaments. End of the fourth phase is uniform concentration and is not shown.
(a) Integral scale of the scalar. The line styles are the same as those in Fig. 6 . The red vertical line indicates ⟨c⟩ = 0.1. (b) Information dimension of the scalar. Line styles and the vertical line are the same as (a).
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(a) Integral scale of the scalar. The line styles are the same as those in Fig. 6 . The red vertical line indicates ⟨c⟩ = 0.1. (b) Information dimension of the scalar. Line styles and the vertical line are the same as (a).
Comparison between simulations and models based on FTLE. Line styles for panels (a) and (b): solid curves: fastest reacting case; dashed curves: slowest reacting case; black curves: DNS results; red curves: modeled from FTLE. (a) ⟨c⟩ against time. (b) ⟨c ^{2}⟩ against ⟨c⟩. (c) Modeled ⟨c⟩ for all realizations at T = 7.5.
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Comparison between simulations and models based on FTLE. Line styles for panels (a) and (b): solid curves: fastest reacting case; dashed curves: slowest reacting case; black curves: DNS results; red curves: modeled from FTLE. (a) ⟨c⟩ against time. (b) ⟨c ^{2}⟩ against ⟨c⟩. (c) Modeled ⟨c⟩ for all realizations at T = 7.5.
(a) Correlation between FTLE w and ⟨c⟩ as a function of time T for different Pe. Black: Pe = 10 000; blue: Pe = 1000: green: Pe = 100; and red: Pe = 10. The scattered points are at T = 7.5, marking the time the largest variation appears in the ⟨c⟩ field. The top inset plot shows the largest gap of ⟨c⟩ between extreme realizations at any given time as a function of Pe. The bottom inset plot shows ⟨c⟩ for Pe = 10 at T = 7.5. (b) Correlation between FTLE w and ⟨c⟩ as a function of time T for different η. Black: η = 4; green: η = 2; blue: η = 1; red: η = 1/2: and magenta: η = 1/4. The scattered points are at the respective times when the variability within each ⟨c⟩ field is the largest. This time scale is inversely proportional to the reaction rate η. The simulations for different η are only up to when ⟨c⟩ gets close to 1 everywhere, hence the correlation beyond this time is extrapolated as the last value of each case (where all curves have started to show asymptotic behavior). The top inset plot shows the largest gap of ⟨c⟩ between extreme realizations at any given time as a function of η. The bottom inset plot shows ⟨c⟩ for η = 4 at T = 1.875.
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(a) Correlation between FTLE w and ⟨c⟩ as a function of time T for different Pe. Black: Pe = 10 000; blue: Pe = 1000: green: Pe = 100; and red: Pe = 10. The scattered points are at T = 7.5, marking the time the largest variation appears in the ⟨c⟩ field. The top inset plot shows the largest gap of ⟨c⟩ between extreme realizations at any given time as a function of Pe. The bottom inset plot shows ⟨c⟩ for Pe = 10 at T = 7.5. (b) Correlation between FTLE w and ⟨c⟩ as a function of time T for different η. Black: η = 4; green: η = 2; blue: η = 1; red: η = 1/2: and magenta: η = 1/4. The scattered points are at the respective times when the variability within each ⟨c⟩ field is the largest. This time scale is inversely proportional to the reaction rate η. The simulations for different η are only up to when ⟨c⟩ gets close to 1 everywhere, hence the correlation beyond this time is extrapolated as the last value of each case (where all curves have started to show asymptotic behavior). The top inset plot shows the largest gap of ⟨c⟩ between extreme realizations at any given time as a function of η. The bottom inset plot shows ⟨c⟩ for η = 4 at T = 1.875.
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Abstract
A study on an advectiondiffusionreaction system is presented. Variability of the reaction process in such a system triggered by a highly localized source is quantified. It is found, for geophysically motivated parameter regimes, that the difference in bulk concentration subject to realizations of different source locations is highly correlated with the local flow topology of the source. Such flow topologies can be highlighted by Lagrangian coherent structures. Reaction is relatively enhanced in regions of strong stretching, and relatively suppressed in regions where vortices are present. In any case, the presence of a divergencefree background flow helps speed up the reaction process, especially when the flow is timedependent. Probability density of various quantities characterizing the reaction processes is also obtained. This reveals the inherent complexity of the reactiondiffusion process subject to nonlinear background stirring.
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