Helicity or twistedness as a fundamental pattern in nature

Engaging with Questions of Higher Order: cognitive vigilance required for higher degrees of twistedness (Part #2)

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As noted below helicity plays a fundamental role at all levels of the organization of matter, whether in the organization of galactic nebula, solar dynamics, dynamics of plasma in nuclear fusion technology, capillary structure (notably in the case of disease), as well as in the very structure and dynamics of DNA in biological cells. The purpose of presenting information here on helicity -- from a seemingly unrelated variety of fields -- is to provide clues for the subsequent discussion of the possible psychological implications of helicity, twistedness and knottedness.

Cosmology and solar dynamics: As noted by L. H. Ford (Twisted scalar and spinor strings in Minkowski spacetime, 1980) with respect to the organization of spacetime, twisted field configurations in Minkowski spacetime are normally associated with a nonsimply connected space. however, it is shown that it is also possible to construct such configurations in a simply connected space.

According to C. R. DeVore (A quantitative accounting of the magnetic helicity released in solar eruptive events. 2000), the helicity, or twistedness, of magnetic fields appears to be a key component of explosive activity on the Sun, manifesting itself in both eruptive prominences and delta sunspot flares. Simulations show the buildup of helicity in the solar corona due to the nonuniform rotation of the Sun, in amounts consistent with those observed to depart in the form of magnetic clouds imbedded in the solar wind.

Electromagnetic fields: Research in space physics, astronomy and astrophysics over the last decade, increasingly reveals the significance of magnetic fields in these areas. These are induced by the motion of ionized matter, known as plasma, which is present in various forms nearly everywhere in the universe. The properties are described by the fluid theory called magnetohydrodynamics that is basic to research on nuclear fusion. As noted in Topological Structure of Electromagnetic Fields in Conducting Fluids:

This interaction of plasma and magnetic field can create an astonishing variety of structures, which often exhibit linked and knotted forms of magnetic flux. In these complex structures of the fields huge amounts of magnetic energy can be stored. It is, however, a typical property of astrophysical plasmas, that the dynamics of magnetic fields is alternating between an ideal motion, where all forms of knottedness and linkage of the field are conserved (topology conservation), and a kind of disruption of the magnetic structure, the so called magnetic reconnection. In the latter the magnetic structure breaks up and re-connects, a process often accompanied by explosive eruptions where enormous amounts of energy are set free.

Magnetic helicity is a function of the vector potential and the magnetic field. It measures the topological linkage of magnetic fluxes. It also manifests itself in the twistedness and knottedness of flux tubes. Through such helicity the linkage of a flux tube with all other flux tubes is preserved when the tubes are filled with infinitely conducting plasma. As noted by J H Hammer (Theoretical aspects of magnetic helicity, 1985):

Even when finite resistivity and magnetic reconnection are allowed, one finds that the global magnetic helicity (accounting for all linkages) is still approximately preserved on the reconnection time scale. Local topological changes (tearing, etc.) that destroy local helicity invariance are allowed as long as the global linkage is preserved.

Hantao Ji. Helicity, Reconnection, and Dynamo. 1999 [text]

Abstract The inter-relationships between magnetic helicity, magnetic reconnection, and dynamo effects are discussed. In laboratory experiments, where two plasmas are driven to merge, the helicity content of each plasma strongly affects the reconnection rate as well as the shape of the difusion region. Conversely, magnetic reconnection events also strongly affect the global helicity, resulting in efficient helicity cancellation (but not dissipation) during counter-helicity reconnection and a finite helicity increase or decrease (but less efficiently than dissipation of magnetic energy) during co-helicity reconnection. Close relationships also exist between magnetic helicity and dynamo effects. The turbulent electromotive force along the mean magnetic field due to either electrostatic turbulence or the electron diamagnetic effect, transports mean-field helicity across space without dissipation. This has been supported by direct measurements of helicity flux in a laboratory plasma. When the dynamo effect is driven by electromagnetic turbulence, helicity in the turbulent eld is converted to mean-field helicity. In all cases, however, dynamo processes conserve total helicity except for a small battery effect, consistent with the observation that the helicity is approximately conserved during magnetic relaxation.

Mark Richard Dennis. Topological Singularities in Wave Fields. 2001 [text]

It is remarkable that, if any wavefield is chosen at random (out of an appropriate ensemble), these singularities occur naturally throughout the field, out of the random interference pattern, and part of the work described here is an exact mathematical calculation of the densities of dislocations in general kinds of random wavefield, as well as the statistical distributions of geometric properties such as curvature, speed (if they are moving) and twistedness. These calculations apply to the threads of silence in a noisy room, or the threads of darkness from light emitted from a thermal radiator (ie a black body).

Bellan Plasma Group of the California Institute of Technology [abstract]

What spheromaks are: Spheromaks are plasmas with very large internal currents and internal magnetic fields that are aligned so as to be in a nearly force-free equilibrium, i.e., the currents are very nearly parallel to the magnetic fields. The spheromak equilibrium is a "natural" state since magnetic turbulence tends to drive magnetically dominated plasmas towards the spheromak state.

Why spheromaks are interesting: Spheromaks are inherently three-dimensional and involve the concept of magnetic helicity which is a measure of the twistedness of a magnetic flux tube. Spheromaks have been proposed as the basis of magnetic fusion confinement schemes and as a means for refueling tokamaks. The physics of spheromaks is closely related to the physics of astrophysical jets.

Meterology: Cyclones, twisters and willy-willies: The largest type of thunderstorm on the planet is known as the supercell. All supercells have an overall rotating structure that contributes heavily to the development of tornadoes, hurricanes and typhoons. They are formed from violently roating giant whirlwinds of air and dense cloud spiraling at over 120 km/hr around a central 'eye' of extreme low pressure. In the United States tornadoes are referred to as twisters and in Australia as willy-willies. They have been termed "the greatest perversion of nature". In the northern hemisphere, hurricane winds circulate around the center in a counter-clockwise fashion.

Geometry and topology: Fundamental to all the above domains in which helicity plays a vital role, is the connection to the branch of mathematics known as topology. The equations describing a simple helix, a coiled-coil and a coiled-coiled-coil, are all of the same essential form.

As noted by Vanessa Robins (Computational Topology at Multiple Resolutions: Foundations and Applications to Fractals and Dynamics, 2000):

Extracting qualitative information from data is a central goal of experimental science. In dynamical systems, for example, the data typically approximate an attractor or other invariant set and knowledge of the structure of these sets increases our understanding of the dynamics. The most qualitative description of an object is in terms of its topology --- whether or not it is connected, and how many and what type of holes it has, for example.

It is in this context that particular torsion coefficients are identified to measure the twistedness of the space in order to provide such qualitiative information.

It is topology that has been able to clarify the nature of helicity and the contraints on twisting and knotting operations that occur in each of the domains. As noted by Jason Cantarella, et al (Influence of Geometry and Topology on Helicity):

The helicity of a smooth vector field defined on a domain in 3-space is the standard measure of the extent to which the field lines wrap and coil around one another; it plays important roles in fluid mechanics, magnetohydrodynamics and plasma physics. In this report we show how the relation between energy and helicity of a vector field is influenced by the geometry and topology of the domain on which it is defined.

Topology has also explored more complex forms of helicity and supercoiling that are not currently identified with any physical phenomena.

Knottedness: The theory of knots is a major area of topology. The genus of a knot is an expression of the degree of "knottedness" of a curve. In geometric topology, genus is the number of holes of a surface. Usually this means the maximum number of disjoint circles that can be drawn on the surface such that the complement is connected. [text] Higher dimensional spaces: knottings inside n-dimensional space have been described by Greg Friedman (Spinning constructions for higher dimensional knots, 2003):

• Simple "spinning" is used to construct 4-dimensional knots from 3-dimensional knots, but can also used to create an n+1 dimensional knot from any n dimen-sional knot.
• Superspinning of classical knots about spheres
• Frame spinning about other manifolds, notabbly to construct inequivalent knots that have the same complement
• Twist spinning, beginning with an n-dimensional knot to obtain an n + 1-dimensional knot
• Frame twist spinning achieved by adding some kind of twisting to the other spins
• Deform spinning including roll spinning
• Frame deform spinning by combining frame spinning and deform spinning

Capillary vessels: The nailfold (the skin overlapping the fingernail at its base) is used in certain forms of medical diagnosis. Certain diseases cause permanent changes to the shapes and densities of nailfold capillaries and therefore nailfold capillaroscopy is important as a tool for diagnosing and monitoring these diseases. For example, B F Jones, et al (A proposed taxonomy for nailfold capillaries based on their morphology) propose a taxonomy for nailfold capillaries that cover six descriptive classes: cuticulis, open, tortuous, crossed, bushy and bizarre. The authors note that earlier studies found that mentally ill patients, and particularly those suffering from schizophrenia, differed from healthy controls in having a decreased number of capillaries and an increased number of bizarre shapes. A standard had been proposed for capillary structure based on their length, thickness (on a scale of 3), twistedness (4 scales) and plexus (5 scales), although rare structures (described as bi-lobed or triple-lobed and stunted) were excluded at that time .

Biological DNA: DNA is a double stranded molecule composed of two polarized strands which run in opposite directions and wind around a central axis. As the double-stranded circular DNA twists around each other they form supercoils -- the axis of the double helix may itself be coiled up in the form of a helix.. This supercoiled DNA contrasts with relaxed DNA. These phenomena will be explored in greater detail below. The DNA's coiled structure expresses a clear magnetic imprint. Replication occurs at radio frequencies, although little is known about DNA's electromeganetic activities.

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