THE SOLAR DYNAMO Mausumi Dikpati High Altitude Observatory, NCAR - - PowerPoint PPT Presentation

High Altitude Observatory (HAO) – National Center for Atmospheric Research (NCAR) The National Center for Atmospheric Research is operated by the University Corporation for Atmospheric Research under sponsorship of the National Science Foundation. An Equal Opportunity/Affirmative Action Employer.

THE SOLAR DYNAMO

Mausumi Dikpati

High Altitude Observatory, NCAR

Organization

• Motivation

(comes from observations)

• Recent Models; Flux-transport Dynamos

(compatible with recent advances in helioseismology)

• Comparison Of Model Output With Observations

(suitable proxies need to be developed from model output)

• Where Are We Now?

(successes; difficulties; possible refinements)

• Summary

(can we predict solar cycle features?)

• Historical Background

(first global solar dynamo models half a century ago)

Manifestations of Solar Activity Cycle

• Appearance and variations

in the number of sunspots with an 11-year periodicity

• Reversal of the Sun’s polar field

after every 11-year

• Large-scale coronal variations

Observed butterfly diagram

• Sunspots are believed to be formed from strong toroidal flux tubes that rise to the

surface due to their magnetic buoyancy

• Equatorward migration of sunspot-belt was explained by an equatorward propagating

dynamo wave for the subsurface toroidal fields

Courtesy: D.H. Hathaway

Historical Background (i) Generation of toroidal field by shearing a pre-existing poloidal field by differential rotation (Ω-effect )

Click to see movie

Historical Background (contd.)

Click to see movie

(ii) Re-generation of poloidal field by lifting and twisting a toroidal flux tube by helical turbulence (α-effect)

Proposed by Parker (1955) Mathematically formulated by Steenbeck, Krause & Radler (1969)

Historical Background (contd.)

• In 1960’s and 70’s, equatorward propagating dynamo wave was obtained by

assuming a radial differential rotation increasing inward throughout the convection zone. Equatorward propagation of dynamo wave was obtained by satisfying Parker-Yoshimura Sign Rule; α dΩ/dr < 0, In North-hemisphere

Observational constraints

(Courtesy: Thierry Corbard)

• But, In 1980’s, helioseismic analysis

inferred that there is no radial shear in the convection zone, and the strong radial shear at or below the base of the convection zone is decreasing inward at sunspot latitudes.

Therefore, Convection Zone Dynamos Do Not Work With Solar-like Ω

More Observational Constraints

Evolution of large-scale, diffuse fields

• Weak diffuse fields drift poleward

in contrast to equatorward migration

• f sunspot belt
• But maintain a 90-deg phase relation

with the sunspots

• Polar reversal takes place during

sunspot maximum

• Polar field changes sign from negative

to positive when subsurface toroidal field is positive

Dikpati & Choudhuri 1995, SolP, 161, 9 [Data source: How ard (NSO) and Wang (NRL)]

Flux-transport Models

Poleward drift of large-scale diffuse fields was explained by invoking a meridional circulation. A θ-Φ surface model by NRL Group in 1989 An r-θ model : Dikpati’s thesis 1996

Click to see movie

What is a Flux-transport Dynamo? ⇒ FLUX-TRANSPORT DYNAMO

(Wang & Sheeley, 1991, ApJ, 375, 761)

Pole

+

Equator

Meridional circulation

1R 0.7R 0.6R

(Choudhuri, Schüssler, & Dikpati, 1995, A&A, 303, L29.) (Durney, 1995, SolP, 160, 213.) (Dikpati & Charbonneau, ApJ, 1999, 518, 508) (Küker, Rüdiger & Schültz, A&A, 2001, 374, 301)

Observational Evidence of Meridional circulation and Differential rotation

Equator

Doppler measurements: Duvall, 1979 Komm, Howard & Harvey 1993 Magnetic tracer : Helioseismic inversions: Helioseismic inversions: Cavallini, Ceppatelli & Righini 1993 Hathaway et al. 1996 Braun & Fan 1998 Hathaway 1996 Giles et al. 1997 Ulrich et al. 1988 Brown et al. 1989 Goode et al. 1991 Tomczyk, Chou & Thompson 1995 Kosovichev 1996 Charbonneau et al. 1997 Corbard et al. 1998

Mathematical Formulation

Under MHD approximation (i.e. electromagnetic variations are nonrelativistic), Maxwell’s equations + generalized Ohm’s law lead to induction equation :

(1)

Applying mean-field theory to (1), we obtain the dynamo equation as,

Differential rotation and meridional circulation Displacing and twisting effect by kinetic helicity Diffusion (turbulent + molecular)

(2)

( ).

B B U B × ∇ − × × ∇ = ∂ ∂ η t

( ),

B B B U B × ∇ − + × × ∇ = ∂ ∂ η α t

Mathematical Formulation (continued)

Under the assumption of axisymmetry, we write; We obtain the following two scalar equations:

(3b) (3a) Toroidal field Poloidal field Meridional circulation Differential rotation

( ) ( )

( ),

ˆ , , ˆ , ,

φ φ φ

t θ r A t θ r B e e B × ∇ + =

( ) ( )

, ˆ , Ω sin ,

φ

θ r θ r θ r e u U + = ( )( )

( )

, 2 2 2

, , sin 1 sin sin 1

φ φ B

B θ r S A θ r η A θ r θ r t A + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − ∇ = ∇ ⋅ + ∂ ∂ u

( ) ( )⎥

⎦ ⎤ ⎢ ⎣ ⎡ ∂ ∂ + ∂ ∂ + ∂ ∂

φ θ φ r

B u θ B ru r r t φ B 1

( )

, 2 2 2

sin 1 ˆ Ω sin

φ φ φ p

B θ r η B η θ r ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − ∇ + × ∇ × ∇ − ∇ ⋅ = e B

Mathematical Formulation (continued)

Babcock 1961, ApJ, 133, 572

Schematic diagram

A Babcock-Leighton type poloidal source-term can be represented as,

( )

( )

. , , 1 erf 1 erf 1 4 1 cos sin , ,

1 2 3 3 2 2 −

⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + × ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − + = B t θ r φ B d r r d r r θ θ S B θ r S

φ

Latitude dependence Amplitude Confines in a thin layer near the surface Quenching

Boundary Conditions Diffusivity profile

B , = = ∂ ∂

φ

θ A

Equator

0.6 R 0.7 R 1 R

Polar Axis

A=0, Bφ=0

A = , Bφ =

( ∇

2

• 1

/ r

2

s i n

2

θ ) A = , B

φ

=

Evolution of Magnetic Fields In a Babcock-Leighton Flux-Transport Dynamo

Click to see movie

Time-latitude Diagrams Produced from the Babcock-Leighton Flux-transport Dynamo Solution

• Equatorward migrating

sunspot belts

• Poleward drifting

large-scale radial fields

• Correct phase relation

between these two fields y diffusivit turbulent source poloidal surface s speed flow max. , 8 . 56

22 . 13 . 89 .

→ → → =

− − T m t m

η υ η s υ T

Equator Equator

t (yr) t (yr)

10 20 30 40 50 10 20 30 40 50 Latitude (degree) Pole Pole Toroidal Field at r = 0.7R Surface Radial Field Latitude (degree)

• Dynamo cycle period (T)

primarily governed by meridional flow speed

Dikpati & Charbonneau, 1999, ApJ, 518, 508

• 1. A Babcock-Leighton dynamo is not self-excited; how can it revive after

Maunder minima?

Difficulties

• 2. Furthermore, N & S

hemispheres are coupled by an antisymmetric magnetic field about the equator, as inferred from Hale’s polarity rule

But, a full-spherical-shell Babcock-Leighton dynamo relaxes to symmetric magnetic fields about the equator

Full Spherical Shell Solutions

Dynamo driven by Babcock-Leighton alpha-effect produces incorrect field symmetry, violating Hale’s polarity rule Dynamo driven by tachocline alpha-effect produces solar-like field symmetry, satisfying Hale’s polarity rule Dikpati & Gilman, 2001, ApJ, 559, 428 Bonanno et al, 2002, A&A, 390, 673

Click to see movie Click to see movie

Summary

• Large-scale solar dynamo mechanism involves 3 basic processes;

(i) Ω-effect, (iii) flux-transport by meridional circulation (ii) α-effect,

• Mean meridional flow sets the solar clock
• Sun is likely to have both Babcock-Leighton type and tachocline α-effect.

“Peculiar” Features Of Cycle 23

19 20 21 22 23

Sunspot Number Monthly Smoothed

1960 1970 1980 1990 2000

Time (years)

250 200 150 100 50

Spot Number

Sunspot index graphics

The monthly (blue) & monthly smoothed (red) sunspot numbers for the latest five cycles

• Rise of this cycle was slow compared to other odd cycles
• It never reached the expected strength
• It showed a second peak during its declining phase,

unusual for an odd cycle

Building A Flux-transport Dynamo-based Prediction Scheme

We postulate that “magnetic persistence”,

• r the duration of the Sun’s “memory” of

its own magnetic field, is controlled by meridional circulation.

Correlation Between Polar Field And Sunspot Field Derived From A Stochastic Flux-transport Dynamo

Charbonneau & Dikpati, 2000, ApJ, 543, 1027 Observationally verified by Hathaway et al, 2002

Polar Field Features Of Cycle 23

Polar field pattern

• Polar reversal in cycle 23

was unusually slow

• After the reversal, polar field

build-up was slow

• S-polar field reversed ~1 yr after

the N-polar field

• S-polar fields were stronger than

N-polar fields during minima of 21 and 22

• During 1993-1999, N- and S-polar

field patterns show distinctly different features

Calibrated Flux-transport Dynamo Model

Dikpati , Corbard, Thompson & Gilman, 2001, ApJ, 575, L41 N-Pole

Green: rotation contours Blue: meridional flow Red: α -effect location

Our supergranular diffusivity value is consistent with that of Wang, Shelley & Lean, ApJ, 2002, 580, 1188

Calibrated dynamo

Click to see movie

Validity test of calibration: Time-latitude diagram to match with observation

Model output Contours: toroidal fields at CZ base Gray-shades: surface radial fields Observed NSO map of longitude-averaged photospheric fields

Effect of time-varying meridional flow (contd.)

• 1. High-latitude reverse cell in N-hemisphere speeds up N-polar reversal

1. Weakening in average active region magnetic flux in cycle 23 slows down polar reversal significantly (matches well with observation)

• 2. Plateau in N-polar field during 1993-1999 well-reproduces the observation.
• 3. However, S-pole reversing ~1/2 yr before N-pole does not match with observation.

Effect of time-varying polar field sources

Combined effect: comparison between model output and observations

• 1. Weak poloidal sources are the

cause of major slow-down in cycle 23 polar reversal

• 2. High-latitude reverse cell in N-

hemisphere is the cause of N- pole reversing before S-pole

• 3. However, S-polar field build-up

is not as slow as observed

(Dikpati, de Toma, Gilman, Arge & White, 2004, ApJ, February 10, in press)

Future Directions: Building a 3D Flux-transport Dynamo

Active longitudes From de Toma, White & Harvey 2000, ApJ, 529, 1101

• Axisymmetric models cannot explain

longitude-dependent solar cycle features.

(Stix, 1971, A&A, 13, 203) (Moss, Touminen & Brandenburg, 1991, A&A, 245, 129)

• Linear studies and nonlinear tachocline

instabilities indicate the existence of m=1 nonaxisymmetry

(Cally, Dikpati & Gilman, 2003, ApJ, 582, 1190) (Gilman & Dikpati, 2000, ApJ, 528, 552)

• First 3D flux-transport dynamo is being

built by incorporating nonaxisymmetry from the tachocline.

(Dikpati, Gilman & van Ballegooijen, under development)