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Non-linear, Topologically Coherent, and Compact Flows Far from Equilibrium

This presentation discusses the theory of Topological Thermodynamics, which explains the emergence of topologically coherent and compact structures in far-from-equilibrium thermodynamic systems. The Pfaff Topological Dimension of a system can dynamically evolve from a turbulent state to an excited topologically "stationary" state. The fundamental topological result relates to the two types of accelerations in fluids: vorticity and convection.

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Non-linear, Topologically Coherent, and Compact Flows Far from Equilibrium

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  1. R.M.K. EGU Vienna 6.06 April 20, 2007 Non-linear, Topologically Coherent, and Compact Flows Far from Equilibrium R. M. Kiehn University of Houston www.cartan.pair.com From the point of view of Continuous Topological Evolution

  2. R.M.K. EGU Vienna 6.06 April 20, 2007 PART I MOTIVATIONS

  3. R.M.K. EGU Vienna 6.06 April 20, 2007 This presentation is a portion of my research interests over the last 40 years. The motivation was based on the recognition thattopological evolution (not geometrical evolution)

  4. R.M.K. EGU Vienna 6.06 April 20, 2007 This presentation is a portion of my research interests over the last 40 years. The motivation was based on the recognition thattopological evolution (not geometrical evolution) is required if non-equilibrium thermodynamic systems and irreversible turbulent processes are to be understood without the use of statistics.

  5. R.M.K. EGU Vienna 6.06 April 20, 2007 The totality of this research is presented as an alternative to the challenge of the Clay Institute, regarding the relationship between Turbulence and the Navier-Stokes equations.

  6. R.M.K. EGU Vienna 6.06 April 20, 2007 The totality of this research is presented as an alternative to the challenge of the Clay Institute, regarding the relationship between Turbulence and the Navier-Stokes equations. The point of departure herein starts with a topological (not statistical) formulation of Thermodynamics, which furnishes a universal foundation for the PDE’s of classical hydrodynamics and electrodynamics.

  7. The theory of Topological Thermodynamics, based upon Continuous Topological Evolutioncan explain why Topologically Coherent, Compact Structures, far from equilibrium, will emerge as long-lived artifacts of Thermodynamic Irreversible, Turbulent, Continuous Processes.

  8. The theory of Topological Thermodynamics, based upon Continuous Topological Evolutioncan explain why Topologically Coherent, Compact Structures, far from equilibrium, will emerge as long-lived artifacts of Thermodynamic Irreversible, Turbulent, Continuous Processes. I want to show that this is due to the fact that: The Pfaff Topological Dimension (PTD) of a Thermodynamic System can dynamically evolve from a turbulent state of PTD = 4 to an excited topologically “stationary” state of PTD = 3.

  9. The theory of Topological Thermodynamics, based upon Continuous Topological Evolutioncan explain why Topologically Coherent, Compact Structures, far from equilibrium, will emerge as long-lived artifacts of Thermodynamic Irreversible, Turbulent, Continuous Processes. I want to show that this is due to the fact that: The Pfaff Topological Dimension (PTD) of a Thermodynamic System can dynamically evolve from a turbulent state of PTD = 4 to an excited topologically “stationary” state of PTD = 3, which is aHamiltonian state far from equilibrium!

  10. The fundamental topological resultExpressed in Engineering Language:There are two types of Accelerations in fluids

  11. The fundamental topological resultExpressed in Engineering Language:There are two types of Accelerations in fluids They are related to rotations and expansions. 1. Vorticity: ω = curl V2. Convection: a = ∂v/∂t +grad V2/2

  12. The fundamental topological resultExpressed in Engineering Language:There are two types of Accelerations in fluids They are related to rotations and expansions. 1. Vorticity: ω = curl V2. Convection: a = ∂v/∂t +grad V2/2 If the Vorticity has a component parallel to the Convective Acceleration, a • ω 0, you have irreversible dissipation, PTD=4, and a 4D Topological Torsion Vector

  13. The fundamental topological resultExpressed in Engineering Language:There are two types of Accelerations in fluids They are related to rotations and expansions. 1. Vorticity: ω = curl V2. Convection: a = ∂v/∂t +grad V2/2 If the Vorticity has a component parallel to the ConvectiveAcceleration, a • ω 0, you have irreversible dissipation, PTD=4, and a 4D Topological Torsion Vector Non-dissipative PTD = 3 Stationary States far from equilibrium imply that a • ω = 0 (a useful design criteria)

  14. Now I am well aware of the fact that THERMODYNAMICS (much less Topological Thermodynamics) is a topic often treated with apprehension.

  15. Now I am well aware of the fact that THERMODYNAMICS (much less Topological Thermodynamics) is a topic often treated with apprehension. In addition, I must confess, that as undergraduates at MIT we used to call the physics course in Thermodynamics, The Hour of Mystery!

  16. Let me present a few quotations that describe the apprehensive views of several very famous scientists: Any mathematician knows it is impossible to understand an elementary course in thermodynamics ....... V. Arnold 1990

  17. Let me present a few quotations that describe the apprehensive views of several very famous scientists: Any mathematician knows it is impossible to understand an elementary course in thermodynamics ....... V. Arnold 1990 It is always emphasized that thermodynamics is concerned with reversible processes and equilibrium states, and that it can have nothing to do with irreversible processes or systems out of equilibrium ......Bridgman 1941

  18. Let me present a few quotations that describe the apprehensive views of several very famous scientists: Any mathematician knows it is impossible to understand an elementary course in thermodynamics ....... V. Arnold 1990 It is always emphasized that thermodynamics is concerned with reversible processes and equilibrium states, and that it can have nothing to do with irreversible processes or systems out of equilibrium ......Bridgman 1941 No one knows what entropy really is, so in a debate (if you use the term entropy) you will always have an advantage ...... Von Neumann (1971)

  19. On the other hand : Einstein, ..., remained convinced throughout his life that thermodynamics is the only universal physical theory that will never be overthrown. .......Uffink 2001

  20. R.M.K. EGU Vienna 6.06 April 20, 2007 I wish to demonstrate that from the point of view of Continuous Topological Evolution, many of the mysteries of Non-Equilibrium Thermodynamics, Irreversible Processes, and Turbulent flows, can be resolved.

  21. R.M.K. EGU Vienna 6.06 April 20, 2007 I wish to demonstrate that from the point of view of Continuous Topological Evolution, many of the mysteries of Non-Equilibrium Thermodynamics, Irreversible Processes, and Turbulent flows, can be resolved. In addition, the non-equilibrium methods can lead to many new processes and patentable devices and concepts.

  22. R.M.K. EGU Vienna 6.06 April 20, 2007 Now, let me present some visual effects that motivated my activity and interests in TOPOLOGICAL THERMODYNAMICS in a period when many scientists and almost no engineers had ever heard about APPLIED TOPOLOGY.

  23. R.M.K. EGU Vienna 6.06 April 20, 2007 Now, let me present some visual effects that motivated my activity and interests in TOPOLOGICAL THERMODYNAMICS in a period when many scientists and almost no engineers had ever heard about APPLIED TOPOLOGY. I will show photos of some compact, topologically coherent objects, that seem to self- organize and appear magically within turbulent, dissipative, environments.

  24. Stimulating Photo #1Long Lived Ionized Ring How can this “long lived” ring form in such a Turbulent Nuclear explosion? (1957)

  25. Stimulating Photo #2Long Lived Scroll Wake How can these scrolls persist without rapid diffusion? 1962 http://www.airtoair.net photo by Paul Bowan

  26. Stimulating Photo #3Jupiter’s Red Spot Why is Jupiter’s red spot so stable?

  27. Stimulating Photo #4Transient Wake formation Wakes are related to topological thermodynamics, Minimal surfaces, Diffraction and Spinors.

  28. Stimulating Photo #5aGalactic Condensation Could the universe be a dilute turbulent gas near its critical point? With its condensate fluctuations as stars and galaxies?

  29. Stimulating Photo #5bHurricane Katrina Can the Spiral wake pattern be a universal topological structure?

  30. Spiral Patterns indicate a non-equilibrium dissipative system of PfaffTopological Dimension 4with irreversible non-Hamiltonian dynamics

  31. Spiral Patterns indicate a non-equilibrium dissipative system of PfaffTopological Dimension 4with irreversible non-Hamiltonian dynamics decaying to a non-equilibrium system of Pfaff Topological Dimension 3,which admits an extremal Hamiltonian dynamics of fixed topology.

  32. Spiral Patterns indicate a non-equilibrium dissipative system of PfaffTopological Dimension 4with irreversible non-Hamiltonian dynamics decaying to a non-equilibrium system of Pfaff Topological Dimension 3,which admits an extremal Hamiltonian dynamics of fixed topology. The acceleration of rotation (angular momentum) is not (topologically) quantized in the PTD=4 state,but becomes quantized in the PTD=3 state.

  33. Spiral Patterns indicate a non-equilibrium dissipative system of PfaffTopological Dimension 4with irreversible non-Hamiltonian dynamics decaying to a non-equilibrium system of Pfaff Topological Dimension 3,which admits an extremal Hamiltonian dynamics of fixed topology. The acceleration of rotation (angular momentum) is not (topologically) quantized in the PTD=4 state,but becomes quantized in the PTD=3 state. Compare to the Bohr atom.

  34. Stimulating Photo #6aLong Lived Falaco Solitons Why do these dimple pairs have such a long lifetime ? 1986 Are they Optical black holes? More later.

  35. Cosmic defects in a Swimming Pool ? Note the “spiral arms” visible about the “black hole” attractors during the formation stages of the stable defect structure.

  36. Stimulating Photo #6bQuarks in a Swimming Pool A macroscopic topological equivalent to quarks on a string 1986 If I have time I will show a movie.

  37. Stimulating Photo #7Cloud Formations Clouds are deformable topologically coherent structures.

  38. What is the Common Thread ? They all are artifacts of : ContinuousTopologicalEvolution creating • Coherent Topological Structures • as Long lived States far from Equilibrium • by means of Irreversible processes.

  39. (CONJECTURE) What is the Common Thread ? They all emerge as Singularities in Finite Time!

  40. I contend that these Universal, Dynamical, Topological Defects, are deformation invariants independent from size and shape.

  41. I contend that these Universal, Dynamical, Topological Defects, are deformation invariants independent from size and shape. They are defined as Topologically Coherent Structures of Pfaff Topological dimension 3 or more. They all exhibit Topological Torsion

  42. I contend that these Universal, Dynamical, Topological Defects, are deformation invariants independent from size and shape. They are defined as Topologically Coherent Structures of Pfaff Topological dimension 3 or more. They all exhibit Topological Torsion Which is a topological artifact of Thermodynamic Non-equilibrium, and exists only in systems of Pfaff Topological Dimension > 2

  43. Stimulating Photo #4Topological Torsion

  44. Topological TorsionWhat is it? Equivalent to a 4D“Torsion field”,T4, dominated by fixed-point accelerations of rotation and expansion in space-time.

  45. Topological TorsionWhat is it? Equivalent to a 4D“Torsion field”,T4, dominated by fixed-point accelerations of rotation and expansion in space-time. The non zero Torsion field, T4≠ 0, appears only in non-equilibrium thermodynamic systems of topological dimension 3 or greater.

  46. Topological TorsionWhat is it? Equivalent to a 4D“Torsion field”,T4, dominated by fixed-point accelerations of rotation and expansion in space-time. The non zero Torsion field, T4≠ 0, appears only in non-equilibrium thermodynamic systems of topological dimension 3 or greater. If the 4-divergence of T4 is not zero, then T4 defines a process direction field which is thermodynamically irreversible.

  47. Topological TorsionHas been ignored by most hydrodynamicists.Pity!

  48. I will show thatIrreversible Processes in the direction of theTopological Torsion Vectorgenerateself similarities,which can be fractal !!!

  49. I will show thatIrreversible Processes in the direction of theTopological Torsion Vectorgenerateself similarities,which can be fractal !!! Are such processes the source of singularity formation in Finite Time ???

  50. Topological Torsion gone berserk Note Spiral Arms, Self-similarities, and Inverse Fractal Dimples In France they eat such things !!! -- a cross between broccoli and cauliflower

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