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8/11/2019 Nandi Lecture

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Dynamo Processes

1st Asia-Pacific School on International Heliophysical Year, Kodaikanal, India 2007

Dibyendu Nandi

Content:• Lecture I: Magnetic Fields in the Universe, Basic MHD

• Lecture II: Dynamo Theory, Formulation, Application to the Sun



Course Outline

Lecture I: Introduction to Magnetic Fields in the Universe

• Galactic, Stellar and Planetary Magnetic Fields• The Sun’s Magnetic Field

• Influence of Solar Magnetic Fields on the Heliosphere• Origin of Astrophysical Magnetic Fields: Dynamo as a Universal Process• Magnetohydrodynamics: Theoretical Concepts

Lecture II: Dynamo Theory – Basic Formulation, Application to the Sun

• Introduction to the Dynamo Problem• An Exercise: Deriving the Mean-Field αΩ Dynamo Equations• Specific Solutions of the Dynamo Equations

• Applications to the Sun: Understanding the Solar Magnetic Cycle• Recent Developments: Simulations of the Solar Dynamo



Lecture I(Monday, 17 th December 2006)



Questions you should ask first…?

• What is a Dynamo?– A process that converts one form of energy into another. E.g., the

electric generator (magnetic + kinetic = electric). Here, we are dealing

with an astrophysical dynamo (electric + kinetic = magnetic field)

• What is Magnetic Field?– A space in which a moving electric charge experiences a non-zero

force. A less stringent, but alternative definition is a space filled with(continuous) magnetic lines of force

• Why Study Dynamo Generation of Magnetic Fields?– Magnetic fields, in situations where they are dynamically

important, play governing roles in many processes such as structureformation in the universe, gamma ray bursts, accretion disks, solar flaresand CMEs – the strongest explosions in the solar system. Therefore it isimportant to understand how these magnetic fields originate and behave



Magnetic Fields in the Universe: From Large to Small Scales

• Astrophysical magnetic fields exist at various scales from those aslarge as intergalactic space, to those comparably smaller on the size ofplanets

• Magnetic fields also exhibit a wide variety of temporal behavior; insome astrophysical systems, they do not vary much on timescalescomparable to the lifetime of that system, while in some they vary

rapidly on very short timescales (e.g., the Sun)

• Their origin and behavior is also different, in different systems;however in general, if in any system the magnetic field varies ontimescales much shorter than the lifetime of the system, then a dynamois probably at the origin

• We will now discuss magnetic fields in astrophysical systems, goingfrom the largest to the smallest scales



Cosmological Magnetic Fields

• On timescales relevant for theevolutionary history of theuniverse, primordial magnetic

fields may have arisen from asecond-order coupling betweenphotons and electrons, mediatedby cosmological density

fluctuations (370,000 yrs afterBig Bang)

• Such a process can give rise tocosmic magnetic fields on theorder of 10 -18 G (1T=10 4 G)at a 1 mega-parsec scale

• Primordial fields can seed dynamoaction in astrophysical systems



Galactic Magnetic Fields

• Galactic structure: Centralcore (bulge), disk and halo

• Magnetic fields theoretically

expected and observed too• Field strength ~ 10 -6 G, i.e.,1/50,000 times that of Earth’s

• This field is ordered and large

scale, permeating the disk • Thought to be produced by

dynamo action• Dynamo amplifies an initial

seed field produced viaSupernova explosionsand shock waves



Galactic Magnetic Fields

• Blue ribbons show simulated magnetic fields and green arrows showobserved field direction, a good match is seen

• Galactic magnetic fields shapes structures through coupling ofcharged (and indirectly dust) particles, plays a role in star formation,and acceleration and the dynamics of Cosmic ray particles



Stellar Magnetic Fields: General (Solar-like)

• Within the class of solar-like stars itself, there is a rich diversity indisplayed magnetic activity, starting from cyclic variations, to steadynon-varying activity to almost no-activity

• Nevertheless some general trends exist; e.g., the magnetic activitylevel correlates very well with the Rossby number (ratio of rotationalperiod to convective turn over time and which is proportional to theinverse square-root of the dynamo number); this implicates the dynamoas the universal mechanism of field generation in these stars



Stellar Magnetic Fields: Extreme (Neutron Stars)

• Collapsed remnant of a massive star (following Supernova explosion)• About 10 Km in radius, very dense, fast rotators (1ms—30s)• Extremely strong magnetic fields ~ 10 12 G• Pulsars: Those neutron stars which emit pulses of radiation• Collapse, field amplification due to flux freezing (dynamo action not

required to explain these strong fields)



Planetary Magnetic Fields (Earth)

• The Earth’s magnetic field is dipolar with strength ∼0.3 G• Has existed for at least 3 billion yrs, however, if it were not continually

generated, would have decayed in about 20,000 yrs• Reversed in sign many times, with average time-span of 200,000 yrs• These clues point to dynamo action, possibly in liquid outer core• Geomagnetic field acts as a protective shield



The Sun’s Magnetic Field: Sunspots

• Appears darker than the surroundings• First telescopic observations by Galileo and Scheiner (early 1600s)• Hale (1908) discovered sunspots are strongly magnetized ~ 1000 G



The Sun’s Magnetic Field: The Solar Cycle

• Number of sunspots observed on the Sun vary with time• Time variation is predominantly cyclic, mean period is 11 years• However, there are large amplitude fluctuations



The Sun’s Magnetic Field: The Butterfly Diagram

• Equatorward migration of sunspots• Poleward migrations of weak surface radial field• Polar field reversal at time of sunspot maximum• Both have an average periodicity of 11 years



The Sun’s Influence on the Heliosphere and Earth

• The Earth and the solar system is immersed in an environment – theheliosphere – which is governed by Sun’s wind and magnetic output



The Sun’s Influence on the Heloisphere

• Solar flares and coronal mass ejections (CMEs), the biggest explosions

in the solar system – eject magnetized plasma and charged particles

• These disrupt: Satellite operations & Telecommunications facilitiesElectrical power grids, Northern oil pipelines

Air-traffic on polar routes



The Sun’s Influence on the Global Climate

• The total solar radiative output (TSI), unfortunately known as the solarconstant until the 1970s, is coupled to the Sun’s magnetic output

• It’s the primary energy input into the global climate system; it varies!• This slow, long-term variation is known to affect global temperature• Maunder minimum – a period of low solar activity (1645-1715 AD) –

coincided with the “Little Ice Age” – a period of global cooling• The Sun’s role in global climate change is still hotly debated



Origin of Astrophysical Magnetic Fields

• Magnetic fields in astrophysical systems – which vary on timescalesmuch shorter than the typical evolutionary timescale of the system –indicates the presence of dynamo action

• Dynamo in a nutshell:

K.E. of Plasma Magnetic Field

(The most basic dynamo? )

• Sources of Kinetic Energy:– Motions of ionized gases & Supernova explosions in galaxies– Motions within the convection zone of stars– Motions of ionized fluids within planetary cores

DYNAMO

An electron in a circular orbit!



Dynamo as an Universal Process: Mathematical Foundations

• Magnetohydrodynamics: The subject that describes the origin anddynamics of magnetic fields in a moving plasma material

• Induction equation:

• The induction equation follows from the Maxwell’s equations and thegeneralized ohm’s law under the non-relativistic approximation

−∇ × E = dB/dt∇ × B = µ J

J = σ (E + V × B)

• Additional assumptions include:– Plasma is a continuum (system scale L >> ion gyro radius)– Plasma is a single fluid (L >> Debye length)– Plasma is in thermodynamic equilibrium

(timescale >> collision timescale, L >> mean-free path)



Theoretical Concepts in MHD: Flux-Freezing

• Governing equation:

• Magnetic Reynolds Number:

• In Astrophysical systems, R M usually high, magnetic fields move withplasma – flux is frozen (Alfven 1942)

2

/ / m

VB L VL R

B Lη η = =



Theoretical Concepts in MHD: The Anti-Dynamo Theorem

• Convective plasma motions may generate magnetic field by induction(Larmor 1919)

• The anti-dynamo theorem (Cowling 1934):

An axisymmetric magnetic field cannot bemaintained by a steady axisymmetric plasma flow

• Rules out dynamo action in systems with certain symmetries in theflow fields

• Can dynamos exist?



Theoretical Concepts in MHD: Around the Anti-Dynamo Problem

• Yes! There are ways around the anti-dynamo theorem

• First positive breakthrough came in 1955, when Eugene Parkerpostulated that helical turbulent motions can play a crucial rolein generating astrophysical magnetic fields through the α -effect

• This is possible because the α -effect removes certain symmetry

constraints on the flow fields

• The α -effect can work only in rotating systems, where Coriolis forceexists and therefore results in non-zero helicity

• Fortunately, most astrophysical systems rotate and also host sustainconvective turbulence!



Theoretical Concepts in MHD: The Mean-Field Alpha Effect

• In a turbulent magnetized medium, the flow and field components canbe expressed as a sum of fluctuating and mean components

• It turns out that the cross-product of the fluctuating components(u` × B`) generates a mean electromotive force (which after somedrastic truncations) can be expressed as:

• Where η is the turbulent diffusivity, α is the mean-field alpha-effect,u` is the fluctuating velocity and τ the correlation time of turbulence

• It is essentially this alpha effect term – related to the Coriolis force –that makes dynamo action possible by bypassing the anti-dynamotheorem



Formulating the Dynamo Problem: The full set of MHD Equations

• The complete set of equations relate to momentum, mass, energyconservation and the induction equation for the magnetic field

• Although a self-consistent approach requires the solution of the fullnumerically, often the kinematic approach is more useful



Deriving the Axisymmetric Kinematic αΩ Dynamo Equtions

• In the kinematic dynamo problem, we solve for the magnetic field witha given velocity field, assuming to direct feedback on the flows

• Axisymmetric Magnetic Fields:

• Axisymmetric Velocity Fields:

• Plug these into the Induction Equation:

• And add the α -effect term to derive the set of kinematic αΩ dynamoequations for the toroidal and poloidal field components



Lecture II(Tuesday, 18 th December 2006)



Sunspots as Tracers of Solar Activity: Tilt & Orientation

• First telescopic observations by Galileo and Scheiner (early 1600s)• Hale (1908) discovered sunspots are strongly magnetized ~ 1000 G• Sunspot pairs have systematic tilt, which increases with latitude• The polarity orientation is opposite in the two hemispheres



Sunspots as Tracers of Solar Activity: The Solar Cycle

• Number of sunspots observed on the Sun vary with time• Time variation is predominantly cyclic, mean period is 11 years• However, there are large amplitude fluctuations



Sunspots as Tracers of Solar Activity: The Butterfly Diagram

• Equatorward migration of sunspots• Poleward migrations of weak surface radial field• Polar field reversal at time of sunspot maximum• Both have an average periodicity of 11 years



Window to the Solar Interior: Plasma Motions

• Interior temperature exceeds a million degrees• Matter exists in the plasma state (highly ionized)• Convection zone has both small scale turbulent flows and large scale

structured flows, in other words we are dealing with…• The dynamics of magnetized plasmas – enter MHD!



Some Issues in MHD: The Governing Equation

• Governing equation:

• Magnetic Reynolds Number:

• In Astrophysical systems, R M usually high, magnetic fields move withplasma – flux is frozen (Alfven, 1942)

• Magnetoconvection (Chandrasekhar 1952, Weiss 1981) – convectiveregion gets separated into non-magnetic and magnetic space – the latterconstitutes flux tubes

2

/ / m

VB L VL R

B Lη η = =



Historical Development – Toroidal Field & Sunspot Creation

Poloidal field Toroidal Field

• Stability – Magnetic Buoyancy (Parker 1955)

ρInternal < ρExternal

• Buoyant eruption, Coriolis forceimparts tilts

π+=

8B

PP2

IE



Historical Development – Toroidal Field Generation –Omega Effect

• Ω-effect in action: Faster rotating equator stretches an poloidal field

in the direction of rotation to create the toroidal fields• Where is the toroidal field generated?

– Convection zone susceptible to buoyancy, ruled out (Parker 1975)– In the overshoot layer, at base of convection zone

(Spiegel & Weiss 1980; van Ballegooijen 1982)



Historical Development – Poloidal Field Generation – The MF α -effect

• Small scale helical convection – Mean-Field α -effect (Parker 1955)• Buoyantly rising 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



Last Two Decade – Flux Tube Dynamics and a Crisis in Dynamo Theory

• Interaction of Coriolis force with buoyantly rising magnetic flux tubeswould force them poleward if B < 10 5 G (Choudhuri & Gilman 1987)

• To match Joy’s law (tilt angles of sunspots) and other morphologicalproperties of solar active regions B ~ 10 5 G (D’Silva & Choudhuri1993, Fan, Fisher & DeLuca 1993)

• Flux tubes with B < 105

G can be stored in the overshoot layer beneaththe base of the convection zone, only stronger flux tubes escape out(Moreno-Insertis, Schüssler & Ferriz-Mas 1992)

• Strength of flux tubes at SCZ base ≈ 10 5 GEquipartition field in convection zone ≈ 10 4 G

• Small-scale helical convection will get quenched – alternative ideasfor poloidal field generation necessary



The Modern Era: Revival of the Babcock-Leighton Idea

• Babcock (1961) & Leighton (1969) idea – decay of tilted bipolarsunspots – distinct from the MF α -effect – and is observed

+ old cycle

– old cycle

h d l l l f l l



The Modern Era: Large Scale Internal Flows from Helioseismology

• Differential rotation in the interior determined from helioseismology,strongest rotational shear at tachocline at the base of the SCZ

• Poleward meridional circulation observed in the outer 15%, massconservation requires a counterflow – possibly near base of the SCZ

l h Ω d l



Formulating the Axisymmetric Kinematic αΩ Dynamo Model

• Axisymmetric Magnetic Fields:

• Axisymmetric Velocity Fields:

• Plug these into the Induction Equation:

to obtain…..

Th Ω D E i



The αΩ Dynamo Equations

• Toroidal field evolution:

• Poloidal field evolution:

• Where the BL alpha effect is: S Bα φ α =

( )( )1 12. sin2 2sin sin

Av r A A S

Pt r r θ η

α θ θ

∂ + ∇ = ∇ − + ∂

( ) ( )

( )22 2

1

1 sin . ( )sin

r

P

Brv B v B

t r r

B r B Br

φ φ θ φ

φ φ

θ

η θ η θ

∂ ∂ ∂ + + ∂ ∂ ∂

= ∇ − + ∇ Ω −∇ × ∇ ×

Model Inputs



Model Inputs

• Observed differential rotation (inferred from helioseismology)• Meridional circulation profile that matches near surface observations• A depth dependent diffusivity profile• A functional form for the BL α -effect (confined to near surface layers)

• Magnetic buoyancy algorithm (transports fields exceeding threshold)

Solar Cycle Simulations



Solar Cycle Simulations(Nandy & Choudhuri 2001, 2002; Nandy 2002, 2003; Nandy 2004a,b;

Wilmot-Smith, Nandy et al 2006; Yeates, Nandy & Mackay 2007)

Toroidal Field Evolution Poloidal Field Evolution





Observations Simulations

What Determines Solar Cycle Amplitude?



What Determines Solar Cycle Amplitude?

If the amplitude of the α -effect is fixed:• Primary constraint: Critical threshold for buoyancy (B c)• Therefore peak toroidal field in the solar interior ~ B c ~ 10 5 G

• Modulation around that by diffusivity and meridional flow

What Determines Solar Cycle Period? Theoretical Simulations



What Determines Solar Cycle Period? Theoretical Simulations

• The speed of the meridional circulation sets the sunspot cycle period• Diffusivity has a small effect

• Note: Period is governed by slowest process in the dynamo chain

What Determines Solar Cycle Period? Observational Confirmation



What Determines Solar Cycle Period? Observational Confirmation

• Based on observed solar cycle properties from 1874-2003 AD• Cycle period is anti-correlated with drift velocity (cross-correlationcoefficient -0.5, confidence level 95%)

• Confirms that the sunspot cycle is driven by meridional flow speed

• This result was the subject of a NASA press release

1.0 1.5 2.0 2.5 3.0Drift Velocity (Degrees/year)

110

120

130

140

150

P e r i o d

( m o n

t h s )

Fluctuations, Memory & Solar Cycle Predictions




• Since magnetic buoyancy, meridional circulation and diffusion all playa role in magnetic flux transport and the latter takes a finite time, itintroduces a memory in the solar cycle (Wilmot-Smith, Nandy, et al.2006)

• Inherent stochastic fluctuations in the dynamo output are natural due tothe turbulent nature of the solar convection zone

• Nevertheless, the memory of these fluctuations will survive based onthe time-delay-dynamics related to magnetic flux transport; thisproperty can be used to make solar cycle predictions

• However, recent attempts lead to conflicting and controversial result(Dikpati et al. 2006, Choudhuri et al. 2007). Why?




, y y(Yeates, Nandy & Mackay 2007)

(Diffusion Dominated Flux Transport) (Advection Dominated Flux Transport)

• Memory of fluctuations different in diffusive and advective regimes• Diffusive flux transport short-circuits advective flux transport• Therefore understanding flux transport key to predictions

What is the Sun’s Role in Global Warming?



g

• Left: Global temperature anomaly courtesy of Mike Mann

(Original source: Mann et al. 2004, Nature, 430, 105)• Right: Solar forcing reconstruction (Nandy and Joy, work in progress)• Yes, the Sun’s activity did rise sharply between 1900-1950• However, the Sun does not seem to be responsible for the continuing

rise in the global temperature since the 1950s

Conclusions: Insights from Solar Dynamo Modeling



g y g

• Using observed large-scale flows (kinematic regime), we can reproducethe observed large-scale magnetic field evolution very well

• So, perhaps we are getting some aspects of the physics right• Flux transport processes such as buoyancy, diffusion and meridional

circulation extremely important for the solar magnetic cycle• Magnetic buoyancy acts as a amplitude limiting factor for solar cycle• Meridional circulation sets the sunspot cycle period• Flux transport mediated time-delay dynamics introduces a memory

mechanism in the Sun, that may be used for predicting the strength offuture cycles

• Development of solar activity predictions is important for satellite

operations, telecommunication facilities, planning of future spacemissions and understanding the future role of the Sun in the context ofglobal climate change

Further Readings



• “Dynamo Models of the Solar Cycle” by Paul Charbonneau, freelyavailable online at: http://solarphysics.livingreviews.org/Articles/lrsp-2005-2/

• “The Physics of Fluids and Plasmas: An Introduction for

Astrophysicists” by Arnab Rai Choudhuri, Cambridge University Press

• Also see research articles and reviews by Dibyendu Nandy at:http://solar.physics.montana.edu/nandi/papers.html

Contact Information



Dibyendu NandiDepartment of PhysicsMontana State UniversityBozeman, MT 50717, USA

Phone: 1-406-994-4470Fax: 1-406-994-4452Email: [email protected]

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