Solar physics meeting at IIA, Bangalore

A possible explanation of
Maunder minimum
from a flux transport dynamo model
Bidya Binay Karak
Department of Physics
Indian Institute of Science

Layout of the talk:
‣Observational features of Maunder
Minimum.
‣Building a flux transport Babcock-Leighton
model.
‣Find out the source of randomness in solar
cycle.
‣Use this model to explain the origin of
Maunder Minimum.

Maunder minimum:
‣Galileo's s Telescope invention in 1609.
‣Maunder minimum period = 1645 to 1715 (Eddy,
1976; Foukal, 1990; Wilson, 1994)
‣68% of days were observed; sunspots appeared
during 2% of the days (Hoyt & Schatten 1996)
‣Related to little ice age.
‣Cosmogenic isotopes: Be 10 and C 14

27 grand minima in last 11,000 years
(from C 14 data by Usoskin et al. 2007)
Do other stars show Maunder Minimum?
Yes (Baliunas et al. 1995).
Characteristics of Maunder Minimum:
period from 1645-1670:
Very few sunspot in two
hemispheres.
period from 1670-1715:
Sunspots mostly on
southern hemisphere
(Ribes & Nesme-Ribes
1993; Sokoloff & Nesme-
Ribes 1994).

Usual 11-year period of
solar activity (Schwab
cycle) was continued.
(Beer et al. 1998,
Miyahara et al. 2004)
1645 1680 1715
years
Sudden initiation but gradual recovery (Usoskin et al.
2000)

• Induction equation:
∂
B
∂
t
= ∇ × ( v × B ) + η ∇ B
2
• Magnetic Reynolds Number:
VB / L
R Rm
= =
ηB/
L
2
VL
η
• In Astrophysical systems, R m usually high, magnetic fields
move with plasma – flux is frozen (Alfven, 1942).
• Magnetoconvection (Chandrasekhar 1952, Weiss 1981) –
convective region gets separated into non-magnetic and
magnetic space – the latter constitutes flux tubes.

Dynamo Model:
Parker (1955)
Axisymmetric magnetic field= toroidal+poloidal
B = B ( r, θ
) e φ
+∇×
[ A ( r, θ
) e
φ
]

Toroidal Field Generation (ω Effect)
(Schou et al. 1998;
Observationally verified Charbonneau et al. 1999)

Poloidal Field Generation:– The Mean Field α-
effect/classical α-effect
• Small scale helical convection – Mean-Field α-effect (Parker 1955)
• Buoyantly yrising toroidal field is twisted by helical turbulent
convection, creating loops in the poloidal plane
• The small-scale loops diffuse to generate a large-scale poloidal field

Towards flux transport dynamo
• Induction equation:
model
∂ B =∇× ( v × B)
+ η ∇
2
B
∂t
t
Evolution of mean (large scale) field is given by
∂ B = ∇× ( v × B + α
B ) + η
2
T
∇
B
∂t
•Our approach is kinematics (velocity field is given)

In axisymmetry case
Velocity field= Ω ( r , θ ) r s in θ e + v re r + v e
Toroidal field evolution:
φ θ θ
differential i rotation meridional circulation
i
B = Br (, θ) eφ
+∇× [ Ar (, θ) eφ]
∂ B
1 ⎡
∂ ∂
⎤
2
1
+ ( rvr B ) + ( vθ
B ) = ηt ( ∇ − ) B + s ( B
p. ∇Ω
)
2
∂t r⎢
⎣∂r ∂θ
⎥
⎦ s
1 ∂ηt
∂
+ ( rB)
r ∂ r ∂ r
Poloidal field evolution:
2
∂A
A
1 1
+ (. v ∇ )( sA) = η (
2
p ∇ − ) A + S
∂t
s s
2
α

Babcock–Leighton alpha effect: (Babcock 1961; Leighton 1969)
S
α
=
α
B
1 ⎡ r−r1 ⎤⎡ r−r2
⎤
0 θ
1 +
( ) 1 ( ) ith 0
⎢
erf
⎥⎢
erf
⎥ 25 /
1 2
α = α cos θ 1 + ( ) 1 − ( ) with = 25 m/s
4⎣ d ⎦⎣ d ⎦
α

Transport mechanisms: 1) meridional circulation 2) diffusivity
where,
and
α
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
10 13
10
12
10 11
η p
, η t
10 10
10 9
0
0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 108
r/R
s
S
v ia θ
s
A s
s

Magnetic buoyancy:
y
Inside flux Tube,
P
out
B
2µ
2
= P
in
+ out in
P
≤
P
Usually the inside is under-dense

Chatterjee et al. (2004)

Fluctuation in polar field:
Joy’s law: Tilts of bipolar
sunspots increase with latitude
Sources of randomness:
Sources of randomness:
1. Fluctuations in the B–L process
of poloidal field generation

2. Fluctuation in the meridional circulation:
Direct/Observation study: (Hathaway 1996;
Snodgrass and Dailey 1996; Ulrich et al. 1988; Basu &
Antia 2000; Haber et al. 2002; Gonz´alez-Hern´andez,
et al. 2006; Gizon & Rempel 2008)
Amplitude can vary from 1 m/s to 100 m/s.

Indirect/Theoretical study: (Hathaway et al. 2003; Javaraiah &
Ulrich 2006; Wang et al. 1991; Dikpati & Charbonneau 1999;
Nandy 2004, Yeates et al. 2008 Wang et al. 2002)
Amplitu ude of meridi ional circulation
Period (yrs)

∂B
1⎡
∂ ∂ ⎤
2 1
+ ( rvrB) + ( vθ
B) = η t( ∇ − ) B + s( Bp. ∇)
Ω
2
∂ t r ⎣
⎢
∂ r ∂
θ
⎦
⎥
s
1 ∂η
t ∂
+ ( rB) ⇒ ∆B ∝ s( Bp. ∇)
Ω∆t
r ∂
r ∂
r
Amplitude of M.C. determines
the strength of toroidal and also
poloidal field
Total suns spot numb ber
Amplitude of meridional circulation

Flux transport dynamo combines three basic
processes:
(i) () strong toroidal field is produced by the stretching of
the poloidal field => involves randomness.
(ii) toroidal field rises due to magnetic buoyancy to
produce sunspots and the decay of tilted bipolar
sunspots produces poloidal field; =>involves
randomness.
(iii) poloidal field diffuses and advects by the
meridional circulation first to high latitudes and then
down to the tachocline.

How to introduce fluctuation?
The simplest way of doing is to change polar field
randomly at each minima of solar cycle.
To reproduce Maunder minimum:
We propose that the polar field at the beginning
of the Maunder minimum fell to a very low value.
1. we stop the code at a solar minimum.
i
2. Change the polar field in the following way:
A N = 00A 0.0A N
A S = 0.4A S
3. we run the code for several cycles.
Choudhuri & Karak (2009)

all
Choudhuri & Karak (2009)

all
Choudhuri & Karak (2009)

1.2
Poloidal field in the solar wind
1
0.8
0.6
0.4
0.2
0
−0.2
1640 1650 1660 1670 1680 1690 1700 1710 1720 1730 1740 1750
Year
Sunspot eruption takes place only when the
toroidal field reaches a value of 10 5 G.
(Choudhuri 1998; D’Silva & Choudhuri 1993;
Fan et al. 1993)
During M.M., Babcock Leighton α effect will
not work but Parker’s traditional mean field α
effect will work.
Choudhuri & Karak (2009)

Dynamo growth rate is
determined by the
dynamo number,
N
D
∝
η
α
2
P
If the duration of the M.M.
is really an indicator of the
dynamo growth time, then
it puts constraints on the
parameters of the model.
Choudhuri & Karak (2009)

Are we entering another grand
minimum right now?
Publication date of our paper:
August, 2009
Choudhuri & Karak (2009)

Are we entering another grand
minimum right now?
No!
The polar field diminution factor γ at the
end of 23 th cycle is 0.6 (Svalgaard,
Cliver & Kamide 2000; Choudhuri,
Chatterjee & Jiang 2007) which is not
sufficient to trigger a Maunder minimum.
Choudhuri & Karak (2009)

Thank You

Path between D to A can be advection dominated or
diffusion dominated (Yeates, Nandy & Mackey 2008)
or can be turbulent pumping p dominated.