Historical Development of Solar
Dynamo Theory

Arnab Rai Choudhuri
Department of Physics
Indian Institute of Science1844: Schwabe discovered solar cycle
1858: Carrington discovered latitudinal drift
1904: Maunder invented butterfly diagram

Hale (1908) discovered magnetic fields in sunspots
Hale et al. (1919) – Often two large sunspots areseen side by side with opposite polarities
A strand of magnetic
flux has come throughthe surface!

Magnetogram map (white +ve, black –ve)
Polarity is opposite (i) between hemispheres; (ii)from one 11-yr cycle to next >> 22-yr period
Tilt of bipolar regions
increases with latitude- Joy’s law (Joy 1919)

Parker (1955) suggested oscillation between the toroidal andpoloidal fields.
Babcock & Babcock (1955) detected the weak poloidal field (~ 10G)

The polar fields and the sunspot number as functions of time
Polar field data from Wilcox Solar Observatory

Babcock and Babcock (1955) – Weak fields outside sunspots(< 10 G)

Unipolar patches shift poleward with solar cycle – Bumba& Howard (1965), Howard & LaBonte (1981), Makarov &Sivaraman (1989), Wang et al. (1989)
Hinode found this field to be concentrated in kilogausspatches (Tsuneta et al. 2008)

Polar field reverses at the time of sunspot maximum.
Weak, diffuse fields must be another manifestation of solar
cycle – ignored by early theorists!

Interaction between velocity & magnetic fields in MHD

For compressible fluids, continuity and energy equations
also have to be solved.
Two approaches to dynamo theory:•Direct numerical simulation (DNS)•Kinematic model - v assumed given, solve only theinduction equation

Early efforts in DNS of solar dynamo by Gilman in 1980s
Glatzmaier & Roberts (1996) successfully simulate thegeodynamo and find geomagnetic reversals

Challenges for DNS of solar dynanmo• Strong statification (density, pressure) withinconvection zone
• Largest relevant scales (106 km) differ from smallest
scales (102 km) by 4 orders

Our theoretical understanding of velocity patterns in theconvection zone (differential rotation, meridional circulation) isvery limited.
Until we can do successful DNS of velocity fields, we have nohope for realistic DNS of the dynamo.
Recent DNS are of exploratory nature and have not reached thestage of detailed comparison with observations (Brandenburg,Nordlund, Brun, Miesch, Toomre)

Kinematic models use the velocity fields discovered byhelioseismology and are able to model many aspects ofobservational data by assigning suitable values to different
parametersInduction equation:

Is it possible to find v and B such that B turns out to beconstant in time?
Cowling’s anti-dynamo theorem (1934) – axisymmetricsolution not possible.
Parker’s turbulent dynamo (1955) – toroidal and poloidal
fields sustain each other.

Axisymmetric magnetic field

Poloidal Component

Toroidal Component

Responsible for weak fields

Gives rise to sunspots

Differential rotation –produces toroidal field frompoloidal field

Parker’s mechanism of generation of poloidal field fromtoroidal field – α-effect

Parker obtained propagating wave solution

Mean field MHD (Steenbeck, Krause & Radler 1966)

Fluctuations around the average
Substituting in induction equation and averaging

whereFirst order smoothing approximation:

where
Dynamo equation

Dynamo equation in spherical geometry
Mangnetic fieldwithVelocity fieldOn substituting in the dynamo equation

T1

T2

T1 >> T2 : αω dynamo (e.g. solar & stellar dynamos)
T2 >> T1 : α2 dynamo(e.g. planetary dynamos)

The αω dynamo equation in rectangular geometry can besolved analytically – model for solar cycle? (Parker 1955)
Steenbeck & Krause (1969)
solved in spherical geometryand obtained the first
theoretical butterfly diagram
Parker-Yoshimura sign rule(Parker 1955; Yoshimura1975) for poleward
propagation

Detailed convection zone dynamo models: Roberts (1972),Kohler (1973), Yoshimura (1975), Stix (1976)

Magnetoconvection
Linear theory – Chandrasekhar 1952
Nonlinear evolution – Weiss 1981; . . .

Sunspots are magnetic field concentrations with suppressed
convection
Magnetic field probably exists as flux tubes within thesolar convection zone

Tachocline – strong differential rotation, generation of toroidal
magnetic field
Flux Tube

Usually the inside is under-dense
Magnetic buoyancy (Parker 1955)
Very destabilizing within
the convection zone, butmuch suppressed below
its bottomDifficulties with convection zone dynamos – Magnetic
buoyancy will remove magnetic flux from convection
zone too quickly (Parker 1975; Moreno-Insertis 1983)
Spiegel & Weiss (1980) and van Ballegooijen (1982)
proposed that the dynamo operates in the overshoot layer
at the base of the convection zone.
This was before helioseismology
discovered the tacholine!!!
Models of interface dynamo –DeLuca & Gilman 1986; Choudhuri
1990; Parker 1993; Charbonneau &MacGregor 1997

3D dynamics of flux tubes in solar convection zone based on
Spruit (1981) thin flux tube equation(Choudhuri & Gilman 1987; Choudhuri 1989; D’Silva &Choudhuri 1993; Fan, Fisher & DeLuca 1993; Caligari et al. 1995)
Early dynamo models
suggested B at bottom to
be 10,000 G, but suchfields are diverted by
Coriolis force(Choudhuri & Gilman
1987)

Only 100,000 G fields can emerge at sunspot latitudes

Joy’s Law (1919) – Tilts of bipolar sunspots increase with latitude

Figure from D’Silva& Choudhuri (1993)– First quantitative
explanation!

Only 100 kG = 100,000 G fields match observations!

Alternative mechanism for poloidal field generation (Babcock1961; Leighton 1969)

Decay of tilted bipolarsunspots
Toroidal field > bipolarsunspots > poloidal field

The original idea of Parker (1955) and Steenbeck, Krause &Radler (1966) involved twisting of toroidal field by helicalturbulence – not possible if toroidal field is 100,000 G.

Magnetic fluxes from active
regions diffuse and getadvected by meridional
circulation

Wang, Nash & Sheeley (1989) –Surface flux transport modelwith actual sunspots as source

Dikpati & Choudhuri (1994) –Flux transport in meridional
plane with interface dynamo
as source

The dynamo cycle - A modified version of the originalidea due to Parker (1955)

Wang, Sheeley & Nash (1991) demonstrated thefeasibility of such a dynamo with a 1D model
Is the condition

satisfied??

Flux transport dynamo in the Sun (Choudhuri, Schussler &Dikpati 1995; Durney 1995)
Differential
rotation > toroidal field
generation
BabcockLeighton process >poloidal field
generation
Meridional circulation
carries toroidal field
equatorward & poloidal
field poleward

From Choudhuri, Schussler & Dikpati (1995) – Meridional
circulation can overcome Parker-Yoshimura sign rule

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…

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)

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

Conclusion
Different phases of solar dynamo models• ~ 1955 – 1980: Convection zone dynamos• ~ 1980 – 1995: Interface dynamo at the tachocline• ~ 1995 – : Flux transport dynamo
The solar cycle is probably produced by a flux transport
dynamo involving the following processes:• Toroidal field generation in tachocline by differential
rotation• Poloidal field generation at surface by Babcock-Leighton
mechanism• Advection by meridional circulation