The Flux Transport Dynamo,
Flux Tubes and Helicity

Arnab Rai Choudhuri
Department of Physics
Indian Institute of ScienceFlux transport dynamo in the Sun (Choudhuri, Schussler & Dikpati 1995;Durney 1995)

Differential rotation >toroidal field generation
Babcock-Leighton
process > poloidal field
generation
Meridional circulation
carries toroidal field
equatorward & poloidal
field poleward

Basic idea was given by Wang, Sheeley & Nash (1991)

Results from detailed model of Chatterjee, Nandy & Choudhuri (2004)
Butterfly diagrams with both sunspot eruptions and weak field at the surface >Reasonable fit between theory & observation

Basic Equations
Magnetic field
Velocity field

The code Surya
solves theseequations

For a range of parameters, the code relaxes to periodic solutions (Nandy &Choudhuri 2002)

Parameters to be specified• Differential rotation(provided byhelioseismology)• Meridional circulation (provided byhelioseismology till depth 0.85R)• Poloidal field source parameter(BL process
observed on the surface, but below the surface?)• Turbulent diffusivities and(surface values
estimated, reasonable assumptions underneath)• Magnetic buoyancyMajor uncertainties at the present time : (i) Penetration depth of
meridional circulation; (ii) Distribution of
below the surface; (iii)
Most satisfactory way of treating magnetic buoyancy.

Constraints onparameters fromobservations• Cyclic behaviour with a period of about 22 yrs• The butterfly diagram of sunspots is restricted to low latitudes(below 40)• The weak fields outside active regions drift poleward• Polar field reversal takes place at the time of sunspot maximum• The solar magnetic field appears dipolar (was it always so?)• Magnetic helicity tends to be negative (positive) in northern(southern) hemisphereFrom Choudhuri, Schussler & Dikpati (1995)

Without meridional circulation

With meridional circulation

Important time scales in the dynamo problem•Ttach – Diffusion time scale in the tachocline•Tconv – Diffusion time scale in convection zone
•Tcirc – Meridional circulation time scale
Ttach > Tcirc > Tconv : Choudhuri, Nandy, Chatterjee, Jiang, Karak, Hotta,
Munoz-Jaramillo ...
Ttach > Tconv > Tcirc : Dikpati, Charbonneau, Gilman, de Toma…

Flux Transport dynamo(Choudhuri, Schussler & Dikpati 1995)

High diffusivity modelLow diffusivity model(diffusion time ~ 5 yrs)

(diffusion time ~ 200 yrs)

IISc group

HAO group(Choudhuri, Nandy,Chatterjee, Jiang, Karak)(Dikpati, Charbonneau,Gilman, de Toma)

Differences between these models were systematicallystudied by Jiang, Chatterjee & Choudhuri (2007) and Yeates,Nandy & Mckay (2008)

Theoretical butterfly diagrams

Dikpati & Charbonneau 1999

Solar-like rotation tends to produce
sunspots at high latitudes > Nonrealistic butterfly diagrams
Kuker, Rudiger & Schultz 2001

Nandy & Choudhuri (2002)
introduced meridional flow
penetrating slightly below thetachocline to produce sunspots atcorrect latitudes

Without penetrating flow

With penetrating flowWe believe that the meridional circulation has to penetrate slightly below
tachocline, but HAO group claim that this is not possible!
The tachocline is the least understood region of the Sun

as function of depth from ourmodel (solid) and from Dikpati &Charbonneau 1999 (dotted)

Gilman & Miesch (2004) argued against the penetration of meriodional
circulation below convection zone, whereas Garaud & Brummel (2008)
found their argument to be flawedDikpati et al. (2004) claim that they can produce good resultswith non-penetrating circulation, but physics details are notclear – no mention of magnetic buoyancy in the paper!!!

No other group could reprocduce their result
Dikpati & Gilman (2008) wrote a strange paper on their method

Distribution of -coefficient (source of poloidal field)
We see the generation of poloidal field at the solar surfaceDikpati & Gilman (2001) and Bonanno et al. (2002) claim that  atthe surface alone would produce quadrupolar parity and argue for in the interior
Chatterjee, Nandy & Choudhuri (2004) show that even surface  canproduce dipolar parity with suitable choice of turbulent diffusion
But we presumably need an  different from Babcock-Leighton  topull the dynamo out of grand minima!

From Chatterjee, Nandy & Choudhuri (2004)

Plots of(solid)and(dashed)
as functions ofdepth(a) Toroidal (b) Poloidal

Hotta & Yokoyama (2010) also obtained dipolar parity by making thediffusivity large near the surface.

Two popular recipes for treating magnetic bouyancy(i)

If B inside convection zone is larger than a critical value, move apart of it to the surface – we follow this(ii) In poloidal field source at surface, put value of B from bottom ofconvection zone – HAO group follow this (first proposed by
Choudhuri & Dikpati 1999)

Munoz-Jaramillo et al. (2011) suggest that the double ring method
originally proposed by Durney (1995) is the best method for treatingmagnetic buoyancy

Magnetic buoyancy is essentially a 3D process and cannot be treatedadequately in a 2D model – main source of uncertainly in 2D flux
transport dynamo models.

Should we go for 3D kinematic models?

Parameters to be specified• Differential rotation(provided byhelioseismology)• Meridional circulation (provided byhelioseismology till depth 0.85R)• Poloidal field source parameter(BL process
observed on the surface, but below the surface?)• Turbulent diffusivities and(surface values
estimated, reasonable assumptions underneath)• Magnetic buoyancy
Major uncertainties at the present time : (i) Penetration depth of
meridional circulation; (ii) Distribution of
below the surface; (iii)
Most satisfactory way of treating magnetic buoyancy.

Dynamo equations just give you information about mean fields
You need to use additional physics to study flux tubes.
How are 105 G flux tubes produced in the tachocline? (Choudhuri 2003)
Toroidal field generation equation:

Toroidal field generated

From which
Order of magnitude

You need to start from at least a few hundred G poloidal field tocreate a 105 G flux tube

From Choudhuri(2003)

Hinode discovered
such flux
concentrations(Tsuneta et al. 2008)

Many sunspots appear twisted

Hale 1927;
Richardson 1941 –Left-handed innorthern hemisphereand right-handed insouthern

Current along the axis of the sunspot

Vector magnetogram measurements show negative
magnetic helicity in northern hemisphere and positive inthe southern (Seehafer 1990; Pevtsov et al. 1995, 2001;
Abramenko et al. 1997; Bao & Zhang 1998).
A quantitative measure of helicity:

From Pevtsov,
Canfield & Metcalf1995

Coronal loop seen by Yohkoh

Martin et al. 1992, 1993 – Coronal filaments also have
opposite polarities in the two hemispheres
Beiber et al. 1987; Smith & Bieber 1993 – Interplanetary
magnetic field also has opposite polarities above & below
equatorial planeWhat gives rise to magnetic helicity?
Magnetic field is generated by dynamo process
Flux tubes rise due to magnetic buoyancy through convection
zone and produce active regions
Longcope, Fisher & Pevtsov (1998) – Helical turbulence incovection zone imparts helicity to rising flux tubes(independent of solar cycle)
Choudhuri (2003); Choudhuri, Chatterjee & Nandy (2004),
Chatterjee, Choudhuri & Petrovay (2006) – Helicity
generation is linked to the dynamo process-effect produces magnetic helicity of the same
sign as 

Babcock-Leighton process => -coefficientconcentrated near the solar surface (positive innorthern hemisphere) => Will it produce positive
helicity in northern hemisphere?

Dynamo equation deals with mean fields, buthelicity is associated with flux tubes!
Choudhuri (2003) studied the connection
between mean field theory and flux tubes

From Choudhuri (2003)
Northern hemisphere
B inside flux tube into
the slide
Poloidal field accreted
around flux tube gives
negative helicity
It is difficult to change magnetic helicity (Woltjer 1958; Taylor1974; Berger 1985)

Dynamo process generates helicity of opposite sign in small andlarge (mean field) scales – Seehafer (1990)
Flux tube to be associated with small scales
At mean field scales, averaging over flux tubes gives positivehelicity

Estimate of magnetic helicity (Choudhuri, Chatterjee& Nandy 2004)

Very simple estimate gives the correct order of magnitude !!!

Helicity calculation from our dynamo model(Choudhuri, Chatterjee & Nandy 2004)

Flux eruption whenever
B > BC above r = 0.71R
 Calculate helicity
Red : positive helicity
Black: negative helicity

Correct helicity during sunspot maxima (negative in north &positive in south)
Helicity reversal at the beginning of a cycle!

Helicity at different latitudes – fromChoudhuri, Chatterjee & Nandy (2004)

Full cycle
Not too bad match with observations!
Fluctuations to be included
During maxima

Beginning of cycle Cycle variation from our model
Cycle variation reported by Bao et al. (2000); Hagino & Sakurai (2005)

Butterfly diagram for helicity

From Zhang et al. (2012)
Observational studies of possible cycle variations of helicity: Bao &Zhang 1998; Hagino & Sakurai 2004; Tiwari et al. 2009; Zhang et al.
2010; Hao & Zhang 2011; Zhang et al. 2012

Build-up of helicity during the rise of the flux tube
through convection zone (Chatterjee, Choudhuri &Petrovay 2006)
Accreted poloidal flux
penetrates inside the flux
tube due to turbulent
diffusion (suppressed insideflux tube)
Sunspot decay by nonlinear
diffusion studied byPetrovay & Moreno-Insertis(1997)

1-D model with radially inward flow!
2-D calculations under progressMagnetic field evolution equations in Lagrangian coordinates:

whereTurbulent diffusivity with magnetic quenching and Kolmogorov
scaling (following Petrovay & Moreno-Insertis 1997):withResults from Chatterjee, Choudhuri & Petrovay (2006)
Magnetic Field falling to low values

Mag. Field restricted above
Thick solid

Solid

DashedDottedDash-dottedMore penetration into flux tube if magnetic field becomes weak in top layers
Detailed comparisons between observation & theory may be possible in future!

Conclusions• The flux transport dynamo explains many aspects of the sunspot cycle,
though there are uncertainties about values of some parameters• Magnetic buoyancy is a 3D process and cannot be included fully
satisfactorily in a 2D model• To produce 105 G magnetic fields inside flux tubes at the base of
convection zone, you need to start from polar field concentrations oforder a few hundred G• Magnetic helicity can be produced by poloidal field getting wrappedaround rising flux tubes• This model predicts a reversal of hemispheric helicyity sign rule at thebeginning of a sunspot cycle