From kinematics to dynamics:
Meridional circulation andtorsional oscillations
Arnab Rai Choudhuri
Department of Physics
Indian Institute of ScienceSpill-over from Lecture 3
First ~15 minutes
Theoretical modelling of grand minima from theflux transport dynamo model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!

Possible mechanisms for producing
grand minima• Fluctuations in Babcock-Leighton process
may make the poloidal field weak
• Fluctuations in meridional circulation may
make it very weak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

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

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

Periods during grand minimum should be longer!

From Choudhuri & Karak (2012)
If poloidal field γ andmeridional circulation v0 at
the beginning of a cycle lie
in the shaded region, thenwe get grand minimum

Durations of last 28 cycles – meridional circulation fluctuations
Strengths of last 28 cycles – polar field fluctuationsintegrated over the shaded area gives
1.3% probability of a cycle getting into grand minima(Choudhuri & Karak 2012)

One of the runs of our dynamo code with fluctuations
We get 20 -30 grand minima in 11,000 years!!!
Recovery mechanism from grand minima poorly understood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

Small scale dynamo - MHD turbulence stretches out seedmagnetic fields until they become strong enough to resist
further stretching
DNS carried out by many groups (Cattaneo 1999; Graham,Danilovic & Schussler 2009)
Comparison between Hinode data and small scale
dynamo DNS

From Graham, Danilovic & Schussler (2009)

Any possible relation between global dynamo and small scale
dynamo?

Time-latitude plot ofmagnetic field outside active
regions => clear contributions
from the global dynamo

From Shiota et al. (2012)
Small flux concentrationsdue to small scale dynamo
and large ones due toglobal dynamo?

Dynamics of differential rotationand meridional circulation
Navier-Stokes equation should be be starting point
Thermodynamics is often important
Turbulence in convection zone has to be handled – mean field
theory?
Classic study by Kippenhahn (1963)
Isotropic viscosity => Solid body rotation
Radial viscosity bigger => equatorial deceleration, poleward
circulation at surface
Radial viscosity smaller => equatorial acceleration,
equatorward circulation at surface

Kitchatinov & Rudiger(1995) calculated theturbulent stresses in theconvection zone from their
turbulence model

Their results on differential rotation and meridional circulation~5O pole-equator temperature
difference needed to drive themeridional circulation

Lorentz force varies periodically with the solar cycle
Does it produce any observable motion?
Kinds of motion expected =>Look at the Navier-Stokes equation

φ-component =>(r, θ)-component =>Periodic variation ofrotation velocity(torsional oscillations)

Periodic variation ofmeridional circulation

Model of torsional oscillations(Chakraborty, Choudhuri & Chatterjee 2009)

Discovered at the surface by Howard & LaBonte (1980)
Helioseismic observations of oscillations within theconvection zone (Kosovichev & Schou 1997; Vorontsov etal. 2002; Basu & Antia 2003; Howe et al. 2005)
Previous theoretical models (Durney 2000; Covas et al. 2000;
Bushby 2006; Rempel 2006) cannot explain the early initiation
at higher latitudes.

Salientobservationalfeatures• The amplitude of oscillation is ~ 5 m s-1 near thesurface ~ 1% of Ω.
• Two branches: Poleward propagating, andequatorward propagating, extending throughout
convection zone.
• Equatorward propagating branch begins 2-3years before the first sunspot eruptions of a cycle
at a higher latitude. Apparent violation ofcausality

Nandy & Choudhuri (2002)
introduced meridional flow
penetrating slightly below thetachocline

Without penetrating flow

With penetrating flow
Strong toroidal fields build up at high latitudes a few years before
sunspot eruptions of the cycle begin

Theoretical Model
Along with the equations of the flux transport dynamo, we
solve the Navier-Stokes equation for vφ

whereis the Lorentz force of the magnetic fieldChoudhuri (2003)
argued on the basisof some dynamo
requirements thatmagnetic field in
the convection zone
should be like thisFirst prediction of strong flux concentrations in the polar regions –later discovered by Hinode (Tsuneta et al. 2008)
Magnetic stresses built up at the based of the convection zone canbe transported upward by Alfven waves (travel time ~ 2 – 3 years)

Comparison between theory and observations

Torsional oscillations at the surface
From Chakraborty, Choudhuri & Chatterjee (2009)

The surface data is from Mount Wilson (Courtesy: R. Ulrich)Comparison between theory and observations
Depth-time plots at latitude of 20o

From Vorontsov et al. (2002)
From our paper

Based on 1996 – 2002 data

Comparison between theory & observations
Time-latitudes plots at different depths

From our paper (0.95R, 0.9R, 0.8R)

From Howe et al. (2005)

From Hathaway &Rightmire (2010) –Variation of meridional
circulation with solar
cycle

Toroidal field at bottom of
convection zone has poleward
Lorentz force (poleward sliptendency)
This opposes meridional circulation
there(Karak & Choudhuri, in preparation)

Conclusions• Grand minima are caused by combined fluctuations inpoloidal field generation and meridional circulation• Small scale dynamo must be operating besides the global
dynamo creating small scale magnetic fields• Hydrodynamics of differential rotation and meridional
circulation is a challenging problem to study through either
DNS or mean field theory• It is more manageable to study the modifications ofdifferential rotation and meridional circulation due to theLorentz force of dynamo-generated magnetic fieldAcknowledgments• My PhD students – Sydney D’Silva, Mausumi Dikpati,Dibyendu Nandy, Piyali Chatterjee, Bidya Karak• Students who did parts of their PhD work with me – Herve
Auffret, Dipankar Banerjee, Jie Jiang, Sagar Chakraborty• My collaborators in solar researcch – Peter Gilman, Aadvan Ballegooijen, Eric Priest, Manfred Schussler, Dana
Longcope, Kristof Petrovay, Jingxiu Wang, . . .
• The person who influenced me most – Gene Parker• Saku Tsuneta for inviting me to NAOJ and for the warm
hospitality