QUIZ Which of these convection patterns is non-Boussinesq?

Homological Characterization Of Convection Patterns Kapilanjan Krishan Marcio Gameiro Michael Schatz Konstantin Mischaikow School of Physics School of Mathematics Georgia Institute of Technology Supported by: DOE, DARPA, NSF

Patterns and Drug Delivery Caffeine in Polyurethane Matrix D. Saylor et al., (U.S. Food and Drug Administration)

Patterns and Strength of Materials Maximal Principal Stresses in Alumina E. Fuller et al., (NIST)

Patterns and Convection Camera Light Source Reduced Rayleigh number  =(T-T c )/ T c  =0.125 Convection cell

Spiral Defect Chaos

Homology Using algebra to determine topology Simplicial Cubical Representations

Elementary Cubes and Chains 0-cube 1-cube 2-cube 1-chain 2-chain 0-chain

Boundary Operator f e1e1 e2e2 e3e3 e4e4 e5e5 e6e6 e7e7 e8e8 dimension # of Loops enclosing holes = of homology group H 1

Homology Summary Patterns are described by Dimension of =, the ith Betti number Homology: Computable topology

Reduction to Binary Representation Hot flow Cold flowExperiment image

Number of Components Zeroth Betti number = 34

Hot flows vs. Cold flows Hot flowCold flow

Spiral Defect Chaos  

 Time ~ 10 3  Number of distinct components Hot flow vs. Cold flow

Number of holes First Betti number = 13

  Time ~ 10 3  Number of distinct holes Hot flow vs. Cold flow

Betti numbers vs Epsilon Hot flow and Cold flow Asymmetry between hot and cold regions Non-Boussinesq effects ? Betti numbers     

Which of these convection patterns is non-Boussinesq?

Simulations (SF 6 ) BoussinesqNon-Boussinesq (Madruga and Riecke)  Q=4.5

Boussinesq Simulations (SF 6 ) Time Series     ComponentsHoles

Non-Boussinesq Simulations (SF 6 )     ComponentsHoles Time Series

Simulations (CO 2 ) at Experimental Conditions     ComponentsHoles  Q=0.7

Boundary Influence

  Time ~ 10 3  Number of connected components Hot flow vs. Cold flow

  Time ~ 10 3  Percentage of connected components Hot flow vs. Cold flow

Convergence to Attractor Frequency of occurrence  cold 0 : Number of cold flow components (  ~1 )

Entropy Joint Probability P(  hot 0,  cold 0,  hot 1,  cold 1 ) Entropy (-  P i log P i )  Bifurcations?

Entropy vs epsilon    Entropy=8.3Entropy=7.9 Entropy=8.9

Space-Time Topology 1-D Gray-Scott model Space Time Time Series—First Betti number Exhbits Chaos

Summary Homology characterizes complex patterns Underlying symmetries detected in data Alternative measure of boundary effects Detects transitions between complex states Space-time topology may reveal new insights Homology source codes available at: