Irregularities of solar cycle and their
theoretical modeling

Arnab Rai Choudhuri
Department of Physics
Indian Institute of Science

Are there regularities longer than the cycles?
The even-odd or Gnevyshev-Ohl rule (Gnevyshev & Ohl 1948) –The odd cycle was stronger than previous even cycle for cycles
10 – 21.

Does the Gleissberg
cycle exist?
Difficult to establish or
refute at present!

Waldmeier effect (Waldmeier 1935)
WE1 – Anti-correlation between cycle strength and rise timeWE2 – Correlation between cycle strength and rise rate(used to predict strength of a cycle)

Dikpati et al. (2008) claimed that
WE1 does not exist in sunspot
area data!
Karak & Choudhuri (2011)
defined rise time as time to growin strength from 20% to 80%

From Karak & Choudhuri (2011)

WE2 is a stronger effect.
WE1 is weaker, but seems to exist!

Correlation between polar field at minimum and next cycle

Choudhuri, Chattejee & Jiang 2007
Jiang, Chatterjee & Choudhuri 2007

Are polar faculae good proxies ofpolar field?
From Sheeley (1991)

From Jiang, Chatterjee& Choudhuri (2007)

Hemispheric asymmetry is never very large• Strengths in two hemispheres (max ~20% in last fewcycles)•Two hemispheres going out of phase (max ~1 year)•Durations of cycles in two hemispheres

Goel & Choudhuri(2009) – Correlation
between asymmetry inpolar faculae at sunspot
minima and aymmetry
in strength of the nextcycleVarious patterns of cycle irregularities• Gnevyshev-Ohl or even-odd effect• Waldmeier effect (rise time or rise rate vs. strength of thecycle)• Correlation between polar field at the minimum and thenext cycle• Hemispheric asymmetry not too large
Possibilities of predicting strengths of future cycles• Polar field at the minimum as a precursor (connected withvarious geomagnetic indices)• Rise rate (after a cycle has begun)

No reliable method for prediction 7-8 years before thesunspot peak!

Polar field during
the last minimum
has been very weak
Schatten (2005) andSvalgaard et al.
(2005) predicted aweak cycle 24

Surprisingly, the first dynamobased prediction by Dikpati &Gilman (2006) is for a verystrong cycle 24!

A fun plot from Pesnell (2008)

Characteristics of Maunder Minimum
Enough observations (Hoyt& Schatten 1994)
Sokoloff & Nesme-Ribes
1994 – Butterfly diagram,hemispheric asymmetry

Usoskin, Mursula & Kovaltsov 2000 – Sudden onset, but gradual
recoveryLess solar activity => weaker B in solar wind => more cosmic
rays => more production of radio-isotopes 10Be and 14C

Beer, Tobias & Weiss 1998 – Results from 10Be ice core data,
solar wind magnetic field continued to have cycles during
Maunder minimum
Miyahara et al. 2004 – Similar conclusion from 14C tree ring
data,cycles within Maunder minimum were longer

History of solar activity before telescopic records reconstructedby Eddy (1977), Stuiver & Braziunas (1989),Voss et al. (1996),Usoskin, Solanki & Kovaltsov (2007)

From Usoskin, Solanki & Kovaltsov (2007) – 27 grandminima and 19 grand maxima in the last 11,000 years!

Are grand minima merely extremes of cycle
irregularities?
Definitive answer not known!

Our view:• Processes which cause cycle irregularities canpush the Sun into grand minima• Recovery from grand minima requires B
generation processes not involving sunspots

Dynamo generation of magnetic fields

Twisting by helicalturbulence – Parker
1955; Steenbeck,Krause & Radler 1966

Decay of tilted bipolarsunspots – Babcock 1961;
Leighton 1969

α–effect – works only if B is not very strong

Early solar dynamo models were mostly based on theα-effectSimulations of flux rise in the rise suggested that B at
the bottom of convection zone is 105 G (Choudhuri
1989; D’Silva & Choudhuri 1993; Fan, Fisher &DeLuca 1993) – α-effect would be suppressed
Flux transport dynamo models are primarily based on
Babcock-Leighton mechanism
Recovery from grand minima must depend again on
the α-effect – spatial distribution or even signunknown!

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

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)

Arguments in favour of high diffusivity model• Diffusivity of order 1012 cm2/s is what you expect frommixing length theory ~(1/3)vl (Parker 1979)• Helps in establishing bipolar parity (Chatterjee, Nandy &Choudhuri 2004; Hotta & Yokoyama 2010)• Keeps the hemispheric asymmetry small (Chatterjee &Choudhuri 2006; Goel & Choudhuri 2009)• Fluctuations spread all over the convection zone soon,explaining many aspects of observations (Jiang, Chatterjee& Choudhuri 2007)• Behaves properly on introducing fluctuations inmeridional circulation (Karak & Choudhuri 2011)

Possible causes of solar cycle
irregularities• Effects of nonlinearities• Fluctuations in poloidal field generation• Fluctuations in meridional circulation

Nonlinearity due to back-reaction of B on vα-quenching:Long history – Stix 1972; Ivanova & Ruzmaikin 1977;
Yoshimura 1978; Schmitt & Schussler 1989
Has a stabilizing effect instead of producing irregularities

Krause & Meinel 1988; Brandenburg et al. 1989 – Nonlinear
stability may detemine the mode of the dynamo
Weiss, Cattaneo & Jones (1984) found chaos in some highly
truncated models with suppression of differential rotation

Charbonneau, St-Jean & Zacharias (2005), Charbonneau,Beaubien & St-Jean (2007) – The Gnevyshev-Ohl rule may be
due to period doubling just beyond bifurcation point

Possible causes of solar cycle
irregularities• Effects of nonlinearities• Fluctuations in poloidal field generation• Fluctuations in meridional circulation

What is the source of fluctuations in poloidal field generation?
Mean field equations obtained by averaging over turbulence
and there must be fluctuations around themFirst suggested by Choudhuri (1992) and explored by Moss etal. (1992) and Hoyng (1993)
Applied to flux transport dynamo by Charbonneau & Dikpati(2000)
Joy’s law: Bipolar sunspots have tilts
increasing with latitude (D’Silva &Choudhuri 1993)
Their decay produces poloidal field(Babcock 1961; Leighton 1969)
Randomness due to large scatter in tilt
angles (caused by convective turbulence– Longcope & Choudhuri 2002)
Supported by Dasi-Espuig et al. (2010)

Cause of correlation between the polar field at theminimum and the next cycle (Jiang, Chatterjee &Choudhuri 2007)
C – Poloidal field produced here byBabcock-Leighton mechanism

P – Field advected to the pole bymeridional circulation
T – Poloidal field diffuses totachocline to produce next cycle

Correlation arises if C -> T diffusion takes 5-10 years (happens
only in high diffusivity model
Prediction of cycle 24 – Dikpati & Gilman (2006) predict a stong
cycle from low diffusivity model and Choudhuri, Chatterjee &Jiang (2007) predict a weak cycle from high diffusivity modelA theoretical mean field dynamo model would producean ‘average’ poloidal field at the end of the cycle.
The actual poloidal field may be stronger or weaker!
We adopt the following procedure(Choudhuri, Chatterjee & Jiang 2007)• The code Surya is run from one
minimum to the next minumum
in the usual way• At minimum we change poloidal
field above 0.8R to match theobserved poloidal field

All calculations are based on our dynamo model (Nandy &Choudhuri 2002; Chatterjee, Nandy & Choudhuri 2004)

Correlations seen in numerical simulations with random
kicks at the sunspot minima (from Jiang, Chatterjee &Choudhuri 2007)

High diffusivity

Low diffusivity(our model)(Dikpati-Gilman)Our final results
for the last
few solar
cycles:(i) Cycles 2123 aremodeled
extremelywell.
(ii) Cycle 24 ispredicted tobe a very
weak cycle!
From Choudhuri, Chatterjee & Jiang (2007)

Production of grand minima
Charboneau, Blais-Laurier & St-Jean (2004) – Low diffusivity
dynamo simulation with fluctuations in α

Intermittencies over several periods – comparable to diffusive
decay time or dynamo memory

Choudhuri & Karak (2009) - Modelling of Maunder minimum
with flux transport dynamo
Assumption : Poloidal field drops to 0.0 and 0.4 of its averagevalue in the two hemispheres

From Choudhuri & Karak (2009)

During recovery poloidal field in solar wind shows
oscillations, even though there are no sunspots
Choudhuri & Karak (2009) also correctly predicted that
we were not entering another grand minimum!!

Possible causes of solar cycle
irregularities• Effects of nonlinearities• Fluctuations in poloidal field generation• Fluctuations in meridional circulation

Period of flux transport dynamo ~ inverse of meridional
circulation speed
Decreases at sunspot maxima(Hathaway & Rightmire 2010)
Due to Lorentz force?
Does not cause irregularities(Karak & Choudhuri 2011)
We disagree with Nandy, MunozJaramillo & Martens (2011)
Possible long-term fluctuations in meridional circulationfrom inverse of cycle durations (Karak & Choudhuri 2011)

Suppose meridional circulation slows down

Dynamo period increases(Yeates, Nandy & Mackay 2008)

Diffusion has moretime to act on the fields
Cycles become weaker

Differential rotation
generates more toroidal
flux
Cycles becomes stronger

Applicable for highdiffusivity dynamo

Applicable for low
diffusivity dynamo

Explanation of Waldmeier Effect (Karak & Choudhuri2011)

WE2 is easy to explain!
But WE1 is reproduced only in high diffusivity model:
weaker meridional circulation causes both longer cycles andweaker cycles.

Karak (2010) matched only the periods of sunspot cycles byvarying meridional circulation, but very surprisingly strengths ofmany cycles also got matched!

Modelling and prediction of actual cycles by using data of polar
fields at minima should work properly only when variations of
meridional circulation has not been large!!!

Karak (2010) found that a sufficiently large decrease in
meridional circulation can cause grand minimum

Periods during grand minimum should be longer!

Conclusions
Various patterns of cycle irregularities• Gnevyshev-Ohl or even-odd effect (explained as nonlinear
bifurcation)• Waldmeier effect (caused by fluctuations of meridional
circulation in high diffusivity model)• Correlation between polar field at the minimum and thenext cycle (caused by fluctuations in poloidal field generation
mechanism in high diffusivity model)• Hemispheric asymmetry not too large (due to diffusive
coupling between hemispheres)Grand minima may be caused by• Fluctuations in poloidal field generation• Fluctuations in meridional circulation
Recovery from grand minima cannot be due to BabcockLeighton mechanism!!