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Helicity and Dynamo theory (Solar Magnetic Fields). Kirill Kuzanyan 1,2 ) 1) IZMIRAN, Russian Academy of Sciences, Moscow, Russia 2) Visiting Professor at National Astronomical Observatories, Chinese Academy of Sciences, Beijing, China.

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Helicity and Dynamo theory (Solar Magnetic Fields)

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Helicity and Dynamo theory(Solar Magnetic Fields)

Kirill Kuzanyan1,2)

1) IZMIRAN, Russian Academy of Sciences, Moscow, Russia

2) Visiting Professor at National Astronomical Observatories, Chinese Academy of Sciences, Beijing, China

Collaboration at National Astronomical Observatories, Chinese Academy of Sciences, Beijing, China

  • Prof. ZHANG, Hongqi

  • Prof. Deng, Yuanyong

  • Dr. Gao, Yu

  • Dr. Xu, Haiqing

  • Dr. Yang, Shangbin

International Collaboration

  • Prof. Dmitry Sokoloff, Moscow

  • Dr. Valery Pipin, Irkutsk, Russia

  • Prof. Nathan Kleeorin & Igor Rogachevskii, Israel

  • Dr. David Moss, Manchester, UK

  • Prof. Axel Brandenburg, NORDITA, Sweden

    Now in Japan (NAOJ, Mitaka):

    Prof. T. Sakurai; Dr. K. Otsuji

The Butterfly Diagram

Basic Physics of the Solar Cycle

  • Solar dynamo theory

    Regeneration of magnetic fields

    due to rotation and

    turbulent convection

  • periodic in time

  • travelling wave

    Parker 1955 dynamo wave

    Babcock & Leighton 1961-69

    Krause & Rädler 1980 mean-field model

Stretch-Twist-Fold Dynamo(after Ya.B.Zeldovich 1960s, also see in H.K.Moffat, 1978)

Generation of mean magnetic fields

Magnetically Linked Spots

Migratory Dynamo wave model

Magnetic field generation

(Parker Dynamo)

E.N. Parker (1955)

Simple Mean field dynamo

  • Parker (1955) was the first to produce quasi-realistic non-axisymmetric velocity distributions with qualitative solutions for B magnetic field

  • The - mean field dynamo theory was introduced by Steenback, Krause & Raedler (1966) and solutions of these equations supported Parker’s picture.

Founders of mean-field dynamo theory

Max Steenbeck, Fritz Krause, Karl-Heinz Raedler

Potsdam, Germany, 1966 - ….

Mean-Field dynamo theory

Turbulent Electro-Motive Force

Simple Dynamo Wave model

Magnetic field generation

(Parker Dynamo)

(Parker 1955)

(A,B): Poloidal/Toroidal field


  • So, the theory requires cyclonic motions (mirror asymmetry of convection), or the alpha-effect to change sign across the equator and be the same from cycle to cycle!

Correlation of Helicities

The Role of Helicities in Dynamo

  • Inviscid integrals

    magnetic helicity

    A.B (for turbulent motion, too!)


    U.B (for classic MHD)

  • Non-linear back reaction in dynamo

    self-consistent models


Observable !

20 years systematic monitoring of the solar vector magnetic fields in active regions taken at Huairou Solar observing station, China (1988-2005…)

More observations from Mitaka (Japan) and also Mees, MSFC (USA) etc.,

but only Huairoudata systematically cover 20 year period!

AR NOAA6619 on 1991-5-11 @ 03:26UT (Huairou)

Photosphetic vector magnetogram

Current helicity over filtergram

Example - Photospheric vector magnetogram of AR 10930 (SOT at Hinode)2006 Dec 11-12 at 23:10:06-00:13:17UT.

AR 10930: HC over the filtergram; positive/negative:0.2, 0.5, 1.0, 4.0 G2/m

Data Reduction

  • 983 active regions; 6630 vector magnetograms observed at Huairou Solar Observing Station;

  • Time average: 2 year bins (1988-2005);

  • Latitudinal average: 7o bins;

    So, each bin contains 30+ magnetograms =>

    => independent statistics in each bin: averages

    with confidence intervals (Student t distribution)

    We assume the data subsamples equivalent to ensembles of turbulent pulsations, so we gather mean quantities in the sense of dynamo theory

Helicity overlaid with butterfly diagram

Twist overlaid with butterfly diagram


Qualitatively, the both helical quantities are distributed in a similar manner. So, despite the noisy nature of the data, accuracy of measurements is reliable.

  • The hemispheric sign rule for helicity:

    Northern hemisphere: mainly negative;

    Southern hemisphere: mainly positive.

    Does not change sign

    from one 11-yr cycle to another!

    While neither current helicity, nor twist are exactly quadratic with magnetic field

Regular inversion of the hemispheric rule:

  • There are particular latitudes and times over the solar cycle at which the hemispheric rule is inverted, mainly at the raise and fall of the 11-yr cycle

  • This invertion is statistically significant ! (cf. Bao et al. 2000, see also Sokoloff et al. 2008)

Helicity plays an important role in the solar dynamo mechanism!

  • Current helicity and twist follow the propagation of the magnetic activity dynamo waves recorded by sunspots and “mean” magnetic fields in active regions

Simple self-consistent dynamo models with evolution of helicity (dynamical nonlinearity)

Kleeorin, Kuzanyan, Moss, Sokoloff, Rogachevskii, Zhang, A&A, 2003;

and a series of publications of the authors thereafter in 2005-2011

Theoretical butterfly diagram for Parker dynamo: helicity in the solar interior (opposite sign than at surface)

Courtesy of D.D. Sokoloff and E. Popova (2009)

[with account of meridional circulation)

Dynamo model with evolution of Helicity

Magnetic field generation

(Parker Dynamo)

Generation of Helicity

Parameterized equation

Development of 1D-2D models(Moss,Kleeorin,Rogachevskii, Sokoloff, Kuzanyan et al.)


Choice of Parameter Range (Example)

C=0.01- ok!



too high value


Kleeorin et al. 2003EVOLUTION OF HELICITY WITH TIME(Northern HemisphereN)




Sunspot Group Number

Magnetic Energy



Diffusion Time Units

Estimate of current helicity of active regions

Zhang, Moss, Sokoloff, Kuzanyan, Kleeorin, Rogachevskii (2012)

2D model with cross-helicity(after Pipin, Kuzanyan, Zhang & Kosovichev, 2011)

Results: model with cross-helicity

  • He

Magnetic field (contour) & Current Helicity (color);

Cross-Helicity (colour) (Kuzanyan, Pipin, Zhang 2007).

Dynamo Model with Total (small-scale + large scale) Magnetic Field Conservation

(Pipin, Sokoloff, Zhang, Kuzanyan 2013 )

Results: model with helicity conservation

  • He

Magnetic field & current helicity: comparing the observations and the model

(Pipin, Zhang, Sokoloff, Kuzanyan, Gao 2013 )

Results: model with helicity conservation

  • He

Magnetic field & current helicity: contour plot

(Pipin, Sokoloff, Zhang, Kuzanyan 2013 )


Thank you!


  • When the night comes with the actionI just know it's time to goCan't resist the strange attractionFrom that giantdynamo....

    ABBA "Summer Night City"

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