THE SOLAR DYNAMO. Mausumi Dikpati High Altitude Observatory, NCAR

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1 THE SOLAR DYNAMO Mausumi Dikpati High Altitude Observatory, NCAR 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.

2 Organization Motivation (comes from observations) Historical Background (first global solar dynamo models half a century ago) 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) Summary (successes; difficulties; possible refinements) Where Are We Now? (can we predict solar cycle features?)

3 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

4 Observed butterfly diagram Courtesy: D.H. Hathaway 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

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

6 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)

7 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

8 Observational constraints 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. (Courtesy: Thierry Corbard) Therefore, Convection Zone Dynamos Do Not Work With Solar-like Ω

9 More Observational Constraints Evolution of large-scale, diffuse fields Weak diffuse fields drift poleward in contrast to equatorward migration of 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: Howard (NSO) and Wang (NRL)]

10 Flux-transport Models Click to see movie 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

11 What is a Flux-transport Dynamo? Pole + Meridional circulation FLUX-TRANSPORT DYNAMO (Wang & Sheeley, 1991, ApJ, 375, 761) (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) 0.6R 0.7R 1R Equator

12 Observational Evidence of Meridional circulation and Differential rotation Doppler measurements: Duvall, 1979 Ulrich et al Hathaway et al Magnetic tracer : Equator Cavallini, Ceppatelli & Righini 1993 Hathaway 1996 Komm, Howard & Harvey 1993 Helioseismic inversions: Giles et al Braun & Fan 1998 Helioseismic inversions: Brown et al Goode et al Tomczyk, Chou & Thompson 1995 Kosovichev 1996 Charbonneau et al Corbard et al. 1998

13 Mathematical Formulation Under MHD approximation (i.e. electromagnetic variations are nonrelativistic), Maxwell s equations + generalized Ohm s law lead to induction equation : B t = ( U B η B). Applying mean-field theory to (1), we obtain the dynamo equation as, B t = ( U B + αb η B), (1) (2) Differential rotation and meridional circulation Displacing and twisting effect by kinetic helicity Diffusion (turbulent + molecular)

14 Mathematical Formulation (continued) Under the assumption of axisymmetry, we write; B ( ), = B ( r θ, t) eˆ + A ( r, θ, t) eˆ U = u( r θ) + r sin θ Ω( r, θ) e, φ, φ φ Toroidal field Poloidal field Meridional circulation We obtain the following two scalar equations: A + t r 1 sin θ Bφ 1 + t r r, ˆφ 2 ( u )( r sin θa) = η A + S ( r, θ, B ) φ B φ, θ ( ru B ) + ( u B ) r φ ( B ) θ φ r 2 1 sin = r sin θ Ω η B eˆ p φ 2 φ θ + η 2 r 2 1 sin 2 Differential rotation B θ φ, (3a) (3b)

15 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 cos sin,, = B t θ r Bφ 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

16 Boundary Conditions Diffusivity profile ( 2-1 /r 2 sin 2 θ)a=0, B φ =0 A=0, B φ =0 Equator 0.6 R 0.7 R 1 R A θ = 0, B = φ 0 Polar Axis A=0, Bφ=0

17 Evolution of Magnetic Fields In a Babcock-Leighton Flux-Transport Dynamo Click to see movie

18 Time-latitude Diagrams Produced from the Babcock-Leighton Flux-transport Dynamo Solution Pole Latitude (degree) Equator Pole Equator Latitude (degree) Toroidal Field at r = 0.7R t (yr) Surface Radial Field t (yr) Dikpati & Charbonneau, 1999, ApJ, 518, 508 Equatorward migrating sunspot belts Poleward drifting large-scale radial fields Correct phase relation between these two fields Dynamo cycle period (T) primarily governed by meridional flow speed T υ s η m 0 T = 56.8 υm s0 ηt, max.flow speed surface poloidal source turbulent diffusivity

19 Difficulties 1. A Babcock-Leighton dynamo is not self-excited; how can it revive after Maunder minima? 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

20 Click to see movie Full Spherical Shell Solutions Dynamo driven by Babcock-Leighton alpha-effect produces incorrect field symmetry, violating Hale s polarity rule Click to see movie 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

21 Summary Large-scale solar dynamo mechanism involves 3 basic processes; (i) Ω-effect, (ii) α-effect, (iii) flux-transport by meridional circulation Mean meridional flow sets the solar clock Sun is likely to have both Babcock-Leighton type and tachocline α-effect.

22 Peculiar Features Of Cycle 23 Sunspot index graphics The monthly (blue) & monthly smoothed (red) sunspot numbers for the latest five cycles 250 Sunspot Number 200 Monthly Smoothed Spot Number Time (years) 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

23 Building A Flux-transport Dynamo-based Prediction Scheme We postulate that magnetic persistence, or the duration of the Sun s memory of its own magnetic field, is controlled by meridional circulation.

24 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

25 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 During , N- and S-polar field patterns show distinctly different features S-polar fields were stronger than N-polar fields during minima of 21 and 22

26 Calibrated Flux-transport Dynamo Model N-Pole Red: α -effect location Green: rotation contours Blue: meridional flow Dikpati, Corbard, Thompson & Gilman, 2001, ApJ, 575, L41 Our supergranular diffusivity value is consistent with that of Wang, Shelley & Lean, ApJ, 2002, 580, 1188

27 Calibrated dynamo Click to see movie

28 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

29 Effect of time-varying meridional flow (contd.) 1. High-latitude reverse cell in N-hemisphere speeds up N-polar reversal

30 Effect of time-varying polar field sources 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 well-reproduces the observation. 3. However, S-pole reversing ~1/2 yr before N-pole does not match with observation.

31 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)

32 Future Directions: Building a 3D Flux-transport Dynamo 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 (Gilman & Dikpati, 2000, ApJ, 528, 552) (Cally, Dikpati & Gilman, 2003, ApJ, 582, 1190) First 3D flux-transport dynamo is being built by incorporating nonaxisymmetry from the tachocline. (Dikpati, Gilman & van Ballegooijen, under development) Active longitudes From de Toma, White & Harvey 2000, ApJ, 529, 1101