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Helicity in Natural Systems: Role, Importance, and Evolution

Explore the role, importance, and evolution of helicity in natural systems, including non-periodic systems, simulations vs observations, and the relationship between helicity and magnetic topology. Learn about helical structures on the Sun, coronal mass ejections, and the conservation laws of helicity. Dive into the assessment of turbulent convective dynamos and the various types of dynamos. Understand the effects of magnetic helicity conservation and its implications in solar wind and MHD experiments.

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Helicity in Natural Systems: Role, Importance, and Evolution

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  1. Role, importance, & evolution of helicity in natural systems • Natural systems? • Nonperiodic systems? • Simulations vs observations • Diven vs decaying turbulence • Dynamo quenching • Other than magnetic helicity? • <A.B>, <J.B>, <w.u>, <u.B>, <y.u>, …

  2. Other names • Twist, swirl, writhe • Handedness, chirality • Torsion • Schraubensinn • Linkage?

  3. Helical structures on the Sun (X-ray emission) Helicity & topology: math. underpinnings Coronal mass ejections

  4. Helical structures on the Sun (X-ray emission) Coronal mass ejections

  5. Moffatt (1969) coined the term in hydro/MHD Coronal mass ejections

  6. Weak force: non-mirrorsymmetric electrons left-handed positrons: right-handed But also spontaneous spin flip unless Electrons have handedness 6x109K

  7. Early universe: conservation law Conseration equation Maximally helical:

  8. Inverse cascading over 8 orders of mag Conseration equation

  9. Sect. 2.2Assessement of the turbulent convective dynamo view

  10. Helicity & linkage • Magnetic helicity: H (B,B) • Current helicity: H (J,J) • Cross helicity: H (w,B) • Kinetic helicity: H (w, w) • Flow helicity: H (u,u)

  11. Stratification & rotation north equator south Cyclones: Down: faster Up: slower No preferred helicity, but + and – possible if result of instability g W g W

  12. Stratification & magnetic field g.B  u.B

  13. Helicity in dynamos • 1966: Steenbeck et al.  kinetic helicity • 1976: Pouquet et al.  kinetic + current helicity • 1982: Kleeorin & Ruzmaikin  quenching • 1992: Vainshtein & Cattaneo catastrophic qu. • 1994: Gruzinov & Diamond  related to helicity • 2000: Blackman & Field  alleviated by fluxes • 2011: solar wind helicity opposite to surface

  14. Recognized in Helicity not mentioned… g W EMF proportional to B

  15. Helicity production + dynamo threshold • LS dynamo threshold • Agreement with helically forced dynamos  zero at the equator? Not necessarily! (Chatterjee et al 2011) Losada et al (2013)

  16.  magnetic helicity • Inverse cascade • (Frisch et al. 1975)

  17. Types of dynamos • Small-scale dynamos • Exponential growth at resistive scale • Sensitive to details or spectrum (Rm dependence) • Large-scale dynamos • a-effect dynamos involve helicity • All others don’t • Negative turbulent diffusivity • Pumping with memory effect • incoh. a-shear effect

  18. Relation to dynamos: feedback & CMEs =coronal mass ejection Blackman & Brandenburg (2003) N-shaped (north) S-shaped (south) (the whole loop corresponds to CME)

  19. Dynamos produce bi-helical fields zero net helicity Magnetic helicity spectrum Southern hemisphere g.W u.w  a.b Pouquet, Frisch, & Leorat (1976)

  20. Two types of twist Finite net helicity (kink unstable) Zero net helicity (alpha effect)

  21. Expect bi-helical fields • Magnetic helicity conserved • Inverse cascade produces small-scale waste! • Opposite sign of helicity (or k) Blackman & Brandenburg (2003)

  22. Resistive effects on inverse transfer

  23. Effects of magnetic helicity conservation Early times: h=0 important Late times: steady state By the time a steady state is reached: net magnetic helicity is generated

  24. Slow-down explained by magnetic helicity conservation molecular value!! Brandenburg: Helical MHD

  25. Vacuum cleaner experiment • Remove small-scale field periodically •  higher saturation field achieved!

  26. Comparison with solar wind data Matthaeus et al. (1982) Measure correlation function In Fourier space, calculate magnetic energy and helicity spectra  Should be done with Ulysses data away from equatorial plane

  27. Measure 2-point correlation tensor u1 u2 Taylor hypothesis: Change of sign: (i) in latitude, (ii) in scale

  28. Comparison • Field in solar wind is clearly bi-helical • ...but not as naively expected Need to compare with direct and mean-field simulations • Recap of dynamo bi-helical fields

  29. Mean-field with dynamical quenching Brandenburg, Candelaresi, Chatterjee (2009, MNRAS 398, 1414)

  30. Helicity in dynamo theory • Magnetic helicity is a conserved quantity • It can only change resistively or via flux! • Need large Reynolds numbers to see the effect in 3-D turbulent dynamo simulations • Most mean-field models ignore the effect • Feedback on alpha effect: term proportional to current helicity <j.b>, but not <J>.<B>

  31. Boundaries instead of periodic Hubbard & Brandenburg, 2010, GAFD

  32. Magnetic helicity flux Gauge-invariant in steady state! • EMF and resistive terms still dominant • Fluxes import at large Rm ~ 1000 • Rm based on kf • Smaller by 2p Del Sordo, Guerrero, Brandenburg (2013, MNRAS 429, 1686)

  33. Magnetic helicity flux • Mostly at large scale!  LS field still Rm dep. Brandenburg (2018, AN 339, 631)

  34. Gauge dependence & robust features • Sign reversal is robust: even for xy & yz sections

  35. 2-point correlation tensor Inspired by earlier 1-D work: Matthaeus et al. (1982) Real space Fourier space  compute Zhang, Brandenburg, & Sokoloff (2014, ApJ 784, L45)

  36. Result for NOAA 11158 Integral scale (inv correlation length) 30,000 G2Mm/(2*6Mm 70,000 G2)=0.04 • Isotropy: EMvert & hor • Positive hel. • Expected for south 30,000 G2Mm x (200Mm)2 = 1043 Mx2/100Mm Zhang, Brandenburg, & Sokoloff (2014, ApJL)

  37. NOAA 11515: wrong sign? negative!! • 18 degr south • but negative hel. • dominance of LS field Zhang, Brandenburg, & Sokoloff (2016, ApJ)

  38. SOLIS data2148-2151 • Positive at small k (large scale) • But total power not strong

  39. Hemispheric sign rule Pevtsov+95 Gosain & Brandenburg (arXiv:1902.11273)

  40. Conclusion • Expect magnetic field to be bi-helical • Small-scale part quenches the dynamo catastrophically (Rm-dependence!) • Need to diagnose magnetic helicity fluxes • Understand magnetic helicity reversal

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