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R. M. Kiehn University of Houston www.cartan.pair.com UNIA Lectures Baeza, Spain June 23-25, 2008 Environmental Turbulence From the new paradigm of Topological Thermodynamics

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UNIA Lectures Baeza, Spain June 22-26, 2008 Problem 1: What is Turbulence ? Problem 2: What is Topological Thermodynamics?

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UNIA Lectures Baeza, Spain June 23-25, 2008 The Bigger problem: How do you define Interaction with the Environment The most general Answer is: In terms of Irreversible processes in Non-Equilibrium Thermodynamic Systems

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UNIA Lectures Baeza, Spain June 23-25, 2008 As we are interested in Environmental Interaction Why Turbulence? The Answer is: Turbulence is a subset of Thermodynamically Irreversible Processes Such processes, that interact with the environment, require Topological Change

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UNIA Lectures Baeza, Spain June 22-26, 2008 What is Turbulence? Unfortunately, No precise (universally accepted) geometrical definition exists. Turbulence (more or less) is (intuitively) a time dependent motion of a continuum fluid which is the antithesis of a streamline flow. The motion may or may not be chaotic or self-similar. However there is a BOTTOM LINE: Almost everyone will agree that Turbulence involves an Irreversible Thermodynamic Process.

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UNIA Lectures Baeza, Spain June 23-25, 2008 My first plan was to use Formal Mathematics to show how certain solutions of the Navier-Stokes Equations V/ t + grad V 2 /2 - V curl V = - dP/ρ + B grad div V - ν 2 V generate Thermodynamically Irreversible Processes and how these solutions may be used to give insight into Turbulence In terms of a universal (topological) dissipation Coefficient, K

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UNIA Lectures Baeza, Spain June 23-25, 2008 For the Navier–Stokes fluid, the Topological Coefficient, K, if NOT zero, insures that flow-process V is thermodynamically irreversible ; equivalently, Turbulent, with a dissipation factor K K = -2(a ω) = 2 {gradP/ρ - B grad div V- ν V 2 V}ω For a Navier Stokes fluid, the universal Dissipation Coefficient, K, is the sum of Baroclinic forces, minus accelerations of expansion (Bulk viscosity) and accelerations of rotation (shear viscosity), times the flow vorticity,. For a Navier Stokes fluid, the universal Dissipation Coefficient, K, is the sum of Baroclinic forces, minus accelerations of expansion (Bulk viscosity) and accelerations of rotation (shear viscosity), times the flow vorticity, ω. For A Plasma K= For A Plasma K= 2(E B) These universal results are detailed in the following related publications: “Turbulence and the Navier-Stokes equations” http:www22.pair.com/csdc/download/clayt7b.pdf http:www22.pair.com/csdc/download/ecosud07.pdf “Prigogine’s Thermodynamic Emergence and Continuous Topological Evolution” http:www22.pair.com/csdc/download/ecosud07.pdf

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UNIA Lectures Baeza, Spain June 23-25, 2008 Part1. What is Topology?

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Prelude: How can an applied scientist, or engineer, use Topology ? Intuitively, Topology is the study of numbers: How many different components of connected parts, how many connected parts contained within the components, how many defects contained within the connected parts, are needed tophysical system. are needed to describe a physical system. Topological Evolution describes how these numbers of component parts change. UNIA Lectures Baeza, Spain June 23-25, 2008

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Prelude: How can an applied scientist, or engineer, use Topology ? Intuitively, Topology is the study of numbers: How many different components of connected parts, how many connected parts contained within the components, how many defects contained within the connected parts, are needed tophysical system. are needed to describe a physical system. Topological Evolution describes how these numbers of component parts change. UNIA Lectures Baeza, Spain June 23-25, 2008

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How can we analyze these problems in turbulent systems when there is not a satisfactory definition in terms of the usual geometric tools of the applied scientist (which are based on diffeomorphic tensor analysis – which preserves existing topology) ? The answer: Use Topological ideas, NOT Geometrical ideas. UNIA Lectures Baeza, Spain June 23-25, 2008

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Topological Evolution is a theory of the emergence, propagation, and condensation, or change in the number of topologically coherent parts (or deformation invariants), which are independent from properties of size or shape. Example: The number of holes in a sheet of rubber is an example of defect structures in the 2d deformable, connected, component. In 3D structures, the connected components are volumes, the 1st defect structures are 2D tubes, and the 2nd defect structures are voids. Holes can be produced by cutting into components or by pasting components together. UNIA Lectures Baeza, Spain June 23-25, 2008

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Consider the sample box (~ representing a physical system) below: Black represents connected components (in the sample box): How many connected components are in the physical sample? The First Betti Number of connected components is a topological property (an invariant of continuous deformation).

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Reconsider the sample box (~ physical system) below: Use different colors to show up the connected components. Now, how many connected components are in the physical sample? The first Betti Number (26) is a topological property ( a deformation invariant) of the sample box (~physical system).

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Now ask: How many connected components in the sample box (~ physical system) have defects (in this case deformed holes)? Use the color red to show up the holes (voids in a connected component). The number of holes (the second Betti number) is apparently 4. The second Betti number is a topological property ( an invariant of continuous deformation) of the sample box (~ the physical system).

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The preceding topological analysis was used to describe time dependent stages of the mechanism of alloy formation in a metallurgical problem. A similar technique has been used to describe the formation of Bernard convection cells in a heated fluid, and the Chaotic evolution of certain chemical reactions. (see http://www.marciogameiro.com/papers/thesis_gameiro.pdf ) http://www.marciogameiro.com/papers/thesis_gameiro.pdf (The evolutionary equations used were phenomenological.) The key idea is that connected parts are topological properties which are invariant under continuous deformation.

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The technique, based upon connectivity, is called Homology Theory, and describes topological properties when the evolution is such that the topology does not change (the Betti Numbers are invariants of continuous deformations). With the evolution of time, the number of connected components and the Betti numbers may change (indicating a topological change of phase or the creation of defects). However, the Homology technique does not yield the Equations of Topological Evolution.

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How do we find the equations of Topological Evolution? The Answer: Use Exterior Differential forms to encode thermodynamic systems. Use Vector direction fields to describe thermodynamic processes. Use the Lie-Cartan differential (not the Covariant Derivative) to describe continuous topological change.

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Homology theory involves the concept of Closure, defined as the union of the interior of a set and its boundary. Co-Homology theory involves the concept of Closure, defined as the union of the set and its limit points. The Two concepts are Dual but not the same. Boundaries and Limit Points are not the same ( necessarily). If the topological sets are defined in terms of Exterior Differential forms, then it can be demonstrated that the Exterior differential is a Limit point Generator.

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UNIA Lectures Baeza, Spain June 22-26, 2008 Topological Evolution of a set is continuous when the Closure of the set on the domain (initial state with Topology T1) evolve, or are included, in the Closure of the image on the range (final state with perhaps a different Topology T2) Topological evolution implies that Topology T2 is not necessarily the same as Topology T1

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UNIA Lectures Baeza, Spain June 22-26, 2008 Topological Evolution Topological Evolution The sets of interest will be exterior differential forms, , of different degrees, as expressed in terms of Cartan’s theory of exterior differential forms. The reason for this choice is that the limit points of an exterior differential form, , can be computed easily in terms of the (exterior) differential process, d limit points of . Topology changes when the # of connected parts changes Pasting parts together is a Continuous Process. Cutting into parts is a Discontinuous Process.

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UNIA Lectures Baeza, Spain June 22-26, 2008 Non-equilibrium Thermodynamics deals with 1. Open thermodynamic systems which can exchange Radiation and Mass with their environment 2. Closed thermodynamic systems which can exchange Radiation with their environment, but not mass. 2. Closed thermodynamic systems which can exchange Radiation with their environment, but not mass. Isolated or Equilibrium thermodynamic systems do NOT exchange radiation or mass with their environment.

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UNIA Lectures Baeza, Spain June 23-25, 2008 Part 2. Topological Thermodynamics and its Topological Thermodynamics and its Universal Results

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UNIA Lectures Baeza, Spain June 23-25, 2008 My current plan is to introduce the fundamental ideas of Topological Thermodynamics and its which are pertinent to My current plan is to introduce the fundamental ideas of Topological Thermodynamics and its Universal Results which are pertinent toTurbulence, Irreversible Processes, and Problems of Non-Equilibrium systems, which involve Environmental Interaction. which involve Environmental Interaction. Secondly I will display several universal visual results of Continuous Topological Evolution that can be used to identify interactions with the environment, and could lead to experimental design techniques for minimizing interaction

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UNIA Lectures Baeza, Spain June 23-25, 2008 Non-Equilibrium Systems which are necessary for the study of interactions with the environment can be studied universally from first principles in terms of Topological Thermodynamics and Continuous Topological Evolution Without Phenomenological Assumptions It turns out that

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UNIA Lectures Baeza, Spain June 23-25, 2008 Continuous Topological Evolution leads to a number of universal ideas and results. 1. The Lie differential process can describe topological change, self similarity and fractal evolution of homogeneous vector spaces. 2. The Lie differential leads to a Cohomological dynamical version of the First Law of Thermodynamics! 3. Topological Torsion is an artifact of non-equilibrium. 4. Irreversible Processes (turbulence) in non-equilibrium systems can cause the emergence of subspaces of Solitons and Wakes 5. There is a universal Topological Dissipation Coefficient, which if non-zero implies the evolutionary process is irreversible.

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UNIA Lectures Baeza, Spain June 23-25, 2008 Continuous Topological Evolution leads to a number of universal ideas and results. 5. Open, Closed, or isolated-equilibrium thermodynamic systems are determined by a Topological (not geometrical) Dimension. 6.The Pfaff Topological Dimension determines the Thermodynamic state. PTD = 4 is an Open non-equilibrium System, PTD = 3 is a Closed non-equilibrium system. 7. PHYSICAL VACUUMS=Vector Spaces of fields of PTD = 4 7a. A Physical Vacuum can have structure. It is not an Empty Void. 7.b A Physical Vacuum of Pfaff Topological Dimension 4 can model a Biological, Economic and Political (or any other) system that interacts with its environment.

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UNIA Lectures Baeza, Spain June 23-25, 2008 Continuous Topological Evolution Permits the development of a theory of Non-equilibrium Thermodynamics, Which is not the same as the classical Equilibrium Thermostatics. The topological evolutionary process is defined in terms of the Lie Differential, which can describe topological change,

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UNIA Lectures Baeza, Spain June 23-25, 2008 Continuous Topological Evolution Assumes vector spaces describing “field properties” can be encoded in terms of Exterior Differential Forms. Typically, a thermodynamic system is encoded in terms of a homogeneous exterior differential 1-form, A = A k (x,y,z,t)dx k - (x,y,z,t)dt. Think (Pressure, Temperature, force, E and B fields, waves, correlations....) Assumes vector spaces describing “particle processes” can be described in terms of Exterior Differential Form Current Densities. Typically, a thermodynamic process is encoded in terms of an exterior differential n-1-form Current density, J = V. Think (Volume, Entropy, Velocity, charge, mole-numbers, collineations...)

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UNIA Lectures Baeza, Spain June 23-25, 2008 The variables of Continuous Topological Evolution are exterior differential forms, that combine the two equivalence classes of vector fields: Recall from matrix Algebra that linear projective vector space methods recognize two evolutionary equivalence classes of Objects: The class of Recall from matrix Algebra that linear projective vector space methods recognize two evolutionary equivalence classes of Objects: The class of Correlations and the Class of Collineations. Similarly, imposition of geometric metric constraints leads to two evolutionary equivalence classes of Objects: The class of Covariant tensors and the class of Contravariant tensors Topological methods based upon continuity and connectivity recognize two evolutionary equivalence classes of Objects: The class of Exterior Differential p-forms and the class of Exterior Differential p-form densities. The class of Exterior Differential p-forms and the class of Exterior Differential p-form densities.

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Continuous Topological Evolution Defines the concept of an evolutionary process in terms of the Lie (topological directional) Differential, NOT the tensor (geometric) Covariant Differential. The Lie differential can describe Topological change, where the Covariant differential cannot. L( V)A= L(J)A = {i(J)d}A+d{i(J)A} UNIA Lectures Baeza, Spain June 23-25, 2008 Based on a Connection Zero

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UNIA Lectures Baeza, Spain June 23-25, 2008 Continuous Topological Evolution (with topological change) is the abstract and dynamical equivalent of the FIRST LAW OF THERMODYNAMICS L(J)A = i(J)dA + d{i(J)A} => Q L(J)A = W + d{U} => Q The Lie differential in the direction, J, of the Action 1- form, A, generates the inexact 1-form of Work, W, plus the differential of the Internal energy, dU, which is equal to the inexact 1-form of Heat = Q. IT is possible to show that relative to the first law of Thermodynamics ALL Tensor directional Covariant Derivatives are ADIABATIC !

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UNIA Lectures Baeza, Spain June 23-25, 2008 I am continuously amazed by the universality of Topological, Non-Equilibrium Thermodynamics The theory generates universal engineering design principles, applicable to Fluids, Electro-Magnetic Plasmas, Cosmological Formulation of Galaxies (as mole condensates of a 4D topological van der Walls gas near its critical point), Superconductivity, Aqueous Ion Chemistry and even (Get This !) gives clues on how to reduce Arthritic Joint Pain!

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UNIA Lectures Baeza, Spain June 23-25, 2008 Recall that Topology is the study of the the number of deformably similar parts, the number of holes,...etc... Topological Non-Equilibrium Thermodynamics Is a study of the number of Components, Phases, Moles, Molecules, Charges,...etc..

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UNIA Lectures Baeza, Spain June 23-25, 2008 Non-Equilibrium Thermodynamics Requires the Pfaff Topological Dimension of a local subspace be greater than 2. PTD > 2 Even though the Geometric Dimension is 4 or greater! PTD = How many independent functions are required to describe the Topology of a Connected subspace. Example: PTD=1 : A sphere in 3D requires only 1 implicit function. PTD < 3 refers to thermodynamic subspaces that are either Isolated or in Equilibrium

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The Pfaff Topological Dimension of the 1-form of Action, A, that encodes a thermodynamic system, PTD(A), determines the thermodynamic equivalence class of system types Open Systems : The Pfaff topological dimension of the Action 1-form, A, must be PTD(A) = 4 Closed Systems: The Pfaff topological dimension of the Action 1-form, A, must be PTD(A) = 3 Isolated-Equilibrium The topological dimension of the Action 1-form, A, is PTD(A) = 2 or less (Pfaff Topological Dimension > 2 implies non-equilibrium) UNIA Lectures Baeza, Spain June 23-25, 2008

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Open Systems : PTD(A) = 4 The system can exchange mole number (mass) and radiation to the environment. Closed Systems: PTD(A) = 3 The system can exchange radiation but not mole number (mass) to the environment. Isolated-Equilibrium PTD(A) = 2 or less These systems can exchange neither radiation nor mole number (mass) to the environment. UNIA Lectures Baeza, Spain June 23-25, 2008

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Part 3. Topological, Non-Equilibrium Thermodynamics And its Turbulent Artifacts

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UNIA Lectures Baeza, Spain June 23-25, 2008 Topological, Non-Equilibrium Thermodynamics Universally involves time dependent Thermodynamically Irreversible Processes Which include TURBULENCE

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Irreversible Processes & Turbulence Are recognized by the topological defects they produce. The Non-Equilibrium Turbulent Fluid is Topologically Disconnected. The isolated-equilibrium fluid is topologically Connected.

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Examples of Turbulence Turbulent recognition in terms of bubble generation and froth. Turbulence recognized by entrapped disconnected bubbles Laminar Non-turbulent

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Turbulence and Topological Defects The topological defects produced by turbulence are often associated with the evolution of deformation invariants, such as a evolutionary change of phase, or the number of parts. Or the condensation from vapor to droplets, or the amalgamation of droplets into liquid, or the creation of wakes, vortices and solitons.

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Turbulence and Topological Defects In this presentation I would like to emphasize a common occurrence, which is the emergence caused by irreversible thermodynamic, turbulent processes, which is the emergence caused by irreversible thermodynamic, turbulent processes, of Spiral Arms and a Tubular Vortex.

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I was puzzled by the persistent long-lived Vortex Ring emergent from the turbulence of a nuclear explosion While detonating Atom Bombs in Nevada

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Note the persistent long-lived Vortex Ring emergent from the turbulence of a nuclear explosion Non- Turbulent region disconnected by phase (mole number of parts) Vortex & Spiral Arm generation by turbulent flows Ex 1.

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Note the persistent long-lived ionized Vortex Ring emergent from the turbulence of a nuclear explosion The Mushroom shape is an indicator of the Rayleigh- Taylor instability. The spiral arms are rotated about z axis, and have Frenet Theory exact solutions Vortex & Spiral Arm generation by turbulent flows Ex 1.

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Vortex & Spiral Arm generation by turbulent flows Ex 2.

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Note the spiral arms in the boundary layer, between the water and air, leading into the water spout vortex core. Limit Cycle Tubular Vortex Vortex & Spiral Arm generation by turbulent flows Ex 2.

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Note the spiral arms in the boundary layer, between the water and air, leading into the water spout vortex core. Vortex and Spiral Arms Generated by Solutions to the equations of Continuous Topological Evolution Limit Cycle core Z = t

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Vortex and Spiral Arms Surprise Two Classes Of Solutions. 1.Those Solutions that decay by contraction from the exterior to the Limit Cycle. 2.A New Solution that decays by contraction from the exterior, Penetrates the Limit cycle, and then expands to the Limit Cycle from the interior. Solution 1Solution 2

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Compare this patented device to the water spout of the previous slide Vortex generation by spiral turbulent flows Tangential entry of warm air creates a Vortex Limit Cycle

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Generate Ubiquitous Logarithmic Spirals But no limit cycles Consider a Dynamical System in a 2D Fluid

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The equations Generate Logarithmic Spirals with Limit Cycles

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Vortex Cores and Spiral Wakes are artifacts of Turbulent Dissipation. Spiral arms generated by turbulent wakes Limit Cycle Core

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The bulk of energy loss for an aircraft is due to the turbulent generation of tip vortices – Save energy-preserve the ecology. Vortex & Spiral Arm generation by turbulent flows

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Wake turbulence is a severe, local, ecological problem. Vortex & Spiral Arm generation by turbulent flows

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Note the persistent long-lived eye of the storm about the vortex core generated in the wake of a C5a Limit Cycle Vortex & Spiral Arm generation by turbulent flows

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Note the spiral arms in the turbulent wake. The Spirals appear to be precursors of the vortex cores. Vortex & Spiral Arm generation by turbulent flows

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The Basic idea is that expansion and rotation are associated with flow fixed points. It can be shown that the 3D dynamical system given be the equations are solutions to the Navier-Stokes Equations in a Rotating Frame of reference, “Some (new) closed form solutions to the Navier-Stokes equations” http:www22.pair.com/csdc/pdf/nvsol2.pdf http:www22.pair.com/csdc/pdf/nvsol2.pdf Vortex & Spiral Arm generation by turbulent flows Ex 3.

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A very simple subset of the formulas are given by the equations of a dynamical system : This dynamical system is a special case of a 3D Tertiary Hopf bifurcation (Langford). B is the expansion parameter Is the rotation parameter This subset of solutions generates the Ubiquitous Logarithmic Spirals. Vortex & Spiral Arm generation by turbulent flows Ex 3.

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Note the similarity between the spiral arm galaxy and the Hurricane. The independence from size is a topological property. Is the Universe mostly a Turbulent Gas ?? With Stars and Galaxies as its defect condensates? Vortex & Spiral Arm generation by turbulent flows Ex 4.

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Hurricane Katrina, generated by turbulent vortex formation, killed or severely damaged 320 million large trees in Gulf Coast forests, which weakened the role the forests play in storing carbon from the atmosphere. The damage has led to these forests releasing large quantities of carbon dioxide into the atmosphere! Ecological Impact of Hurricane Katrina Spiral arms with Limit cycle core

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The generation of a Super Cell leading to Tornados Cloud and Vortex generation by turbulence

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Super Cell from space

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Super Cell Section

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Dissipative turbulent generation of a persistent vortex. Ecological damage can be enormous. Vortex & Spiral Arm generation by turbulent flows Ex 5. Vortex Core

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Cold -30 Hot 100 Ranque-Hilsch tube ALSO Checkout the Windhexe machine at http://www.youtube.com/watch?v=xxuM7xWL5RQ Vortex & Spiral Arm generation practical Apps

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UNIA Lectures Baeza, Spain June 22-26, 2008 Turbulence and its Visual Artifacts Spiral Arms, Vortices, Wakes and Solitons Are Residues of Turbulent Decay, Which have EMERGED from a Dissipative Open Thermodynamic Systems Producing subsets of Closed Non-Equilibrium Thermodynamic Systems Summary of

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UNIA Lectures Baeza, Spain June 22-26, 2008 Expansions and Rotations are associated with Spiral Arms But a Topological Dimension of 3 is required for Irreducible Thermodynamic Dissipation (hence 2D Turbulence is a Myth.) The Existence of the Property of TOPOLOGICAL TORSION Insures the topological dimension is 3 or more. In a dynamical system

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Yet, Almost NO Engineers, and Very Few Physicists Understand TOPOLOGICAL TORSION (pity) Yet Topological Torsion is a key design tool for controlling and understanding non-statistical TURBULENCE Topological Torsion and Topological Spin arXiv:physics/0102001 arXiv:physics/0102001

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For a Navier Stokes fluid (with PTD=4) the 4-component Topological Torsion Vector, T 4 = [T, h], is deduced as: When the expression for acceleration is substituted : T4= [T, h]=0.T4 If PTD <3, then T4= [T, h]=0. T4 is an artifact of non-equilibrium !!

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Accelerations of Expansion (with non Zero divergence and Bulk Viscosity) and Accelerations of Rotation (with rotations and Shear Viscosity) Coupled with Baroclinic forces (per mole) if not Orthogonal to the Vorticity, will generate Continuous Irreducible Thermodynamic Dissipation In a Fluid, the sum of K = -2(a ω) = 2 {gradP/ρ - B grad div V- ν V 2 V}ω

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Frenet Equations, Spinors and Wakes Using ideas generated by Continuous Topological Evolution Click here to run a QBasic 1989 program The unit tangent Vector, t The unit normal Vector n Spiral shapes

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Frenet Equations, Spinors and Wakes Using ideas generated by Continuous Topological Evolution Click here to run a QBasic 1989 program Flow past a sharp edge

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Summary of Continuous Topological Evolution with Lessons from Topological Thermodynamics.

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Axioms of Topological Thermodynamics 1. Topological Structure of Physical Systems is encoded by an Action differential 1-form A. 2. Physical Processes can be defined in terms of contravariant Vector/Spinor direction fields, V. 3. Continuous Topological Evolution is encoded by Cartan's magic formula ~ First Law L (V) A = i(V)dA+di(V)A = W + dU = Q 4. Pfaff Topological Dimension, PTD, classifies different categories of Systems and Processes

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The most important properties are 1. The Pfaff Topological Dimension - PTD(A) Or class of a 1-form, A. Topological properties of Thermodynamics from Exterior Differential forms The PTD of a 1-form A is equal to the number of non-zero terms in the Pfaff Sequence {A,dA,A^dA,dA^dA…}. The PTD, or class of a 1-form, is equal to the MINIMUM number of functions (of base variables – or coordinates) required to express the topological features of the connected component.

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Thermodynamic Systems can be put into topological equivalence classes determined by the PTD(A) of the 1-form of Action. Open Systems : PTD(A) = 4 Closed Systems: PTD(A) = 3 Isolated-Equilibrium Systems: PTD(A) 2 (Pfaff Topological Dimension > 2 implies non-equilibrium) The Pfaff Topological Dimension of A

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Open : PTD(A) = 4 The system can exchange mole number (mass) and radiation to the environment. Closed: PTD(A) = 3 The system can exchange radiation but not mole number (mass) to the environment. Isolated-Equilibrium systems, PTD(A) = 2 or less Do NOT interact with the environment. The Pfaff Topological Topological Interaction with the Environment

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Based on the Pfaff Topological Dimension of W Thermodynamic Reversible Processes imply that the Heat 1-form, Q is integrable. Q^dQ = 0 Extremal:PTD(W) = 0, W = 0, Hamiltonian Bernoulli-Casimir: PTD(W) = 1, W exact Hamiltonian Helmholtz: PTD(W) = 1, W closed Conservation of Vorticity Each of these flows are thermodynamically reversible, as dW = 0 = dQ, implies Q ^ dQ = 0.

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All of the above processes are Reversible! These theories make up the bulk of Classical Dynamics!! THESE PROCESSES (and theories) CAN NOT DESCRIBE TURBULENCE OR INTERACTION with the ENVIRONMENT To Be Irreversible the process and the thermodynamic system must be such that Q^dQ is not zero The Topological Torsion (based on Q) is NOT ZERO. PTD(Q) > 2.

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www.cartan.pair.com

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Spinor Surfaces of Zero Mean Curvature WHEELER WORM HOLE Euclidean signature. Gauss Curvature < 0 FALACO SOLITON Minkowski signature. Gauss Curvature > 0 Minimal Surface Connected surface Open Throat Maximal Surface Pair Singular thread between Disconnected Vertices Heresy #6. Topological Coherent states (Solitons) can exist in non equilibrium systems

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“Turbulence and the Navier-Stokes equations” http:www22.pair.com/csdc/download/clayt7b.pdf http:www22.pair.com/csdc/download/clayt7b.pdf http:www22.pair.com/csdc/download/ecosud08.pdf “Prigogine’s Thermodynamic Emergence and Continuous Topological Evolution” http:www22.pair.com/csdc/download/ecosud08.pdf Copies of the entire ppt program can be found at http://www22.pair.com/csdc/download/unia.ppt The basic files are to be found at http://www22.pair.com/csdc/download/homoclin.exe http://www22.pair.com/csdc/download/splate.exe

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