1. THE CIRCULATION DOMINATED SOLAR DYNAMO MODEL REVISITED Gustavo A. Guerrero E. (IAG/USP) Elisabete M. de Gouveia Dal Pino (IAG/USP) Jose D. Muñoz (UNAL) IV WORKSHOP NOVA FISICA NO ESPACIO, CAMPOS DO JORDAO, FEVEREIRO 20-25

2. Contents Sunspot cycle The Solar dynamo Mathematical Formalism (MHD) Results Conclusions

3. SUNSPOT CYCLE FIGURES Pairs of sunspots appearance, tilt of inclination (Joy’s law). Latitude of appearance (Spörer`s Law) Inversion of polarities, 11 years cycle (Hale`s Law) Equatorward migration of sunspots Intensity of the Magnetic Fields ~10 3 G for the sunspots Tens of G. for the diffuse poloidal field Sunspots pair Observational butterfly diagram (NASA) Solar cycle

4. THE SOLAR DYNAMO 1. Dipolar initial magnetic field 2. Differential rotation -> belt of toroidal field arround the solar equator. 3. Magnetic Flux tubes (ρi Magnetic bouyancy. 4. Tilt bipolar active magnetic regions (BMR) 5. Decay of BMR`s. (Fan et al. 2003)

5. MATHEMATICAL FORMALISM The MHD induction equation is: By assuming spherical symmetry: Differential Rotation (1) (2) (3) Replacing (2) and (3) in (1) and separating the poloidal and toroidal components, we obtain: (4) (5) Meridional Circulation

6. Differential rotation Helioseismologic observations: dΩ/dr(r=0.7 Rסּ) (6) (Schou et al., 1998)

7. MATHEMATICAL FORMALISM The MHD induction equation is: By assuming spherical symmetry: Differential Rotation (1) (2) (3) Replacing (2) and (3) in (1) and separating the poloidal and toroidal components, we obtain: (4) (5) Meridional Circulation

8. A superficial flow about 20m/s is observed in all latitudes, but the way how the counter flow happens is until now unknown. We assume a simple convection cell by meridional quadrant thus: (Haber et al., 1998) (7) (8)

9. MATHEMATICAL FORMALISM The MHD induction equation is: By assuming spherical symmetry: Differential Rotation (1) (2) (3) Replacing (2) and (3) in (1) and separating the poloidal and toroidal components, we obtain: (4) (5) Meridional Circulation

10. Magnetic Bouyancy Diffusivity Magnetic diffusivity for the toroidal field Magnetic diffusivity for the poloidal field (9) (10) (11)

11. MATHEMATICAL FORMALISM The MHD induction equation is: By assuming spherical symmetry: Differential Rotation (1) (2) (3) Replacing (2) and (3) in (1) and separating the poloidal and toroidal components, we obtain: (4) (5) Meridional Circulation

12. TWO IMPORTANT MODELS Dikpati & Charbonneau, 1999, ApJ, 518, 508. –Solar like differential rotation. –Non local source formulation. Nandy & Choudhuri, 2002, Science,296, 1671. –Deep Meridional Flow. –Numerical formulation of the source term.

13. RESULTS ParameterValue Uo2000 cm/s So25 cm/s ηCηC 2.4x10 11 Rb (deep)0.675 R סּ We solved the equations (4) and (5) inequations (4) and (5) a two dimensional mesh of 128x128 spatial divisions, with 0.55<r<1 Rסּ, and 0<θ<π/2 and the next boundary conditions: At θ=0A=0 B=0 At θ=π/2 dA/dt=0B=0 At r=0.55A=0B=0 At r= Rסּ (free space cond)B=0 Toroidal field Radial field

14. Deep meridional Flow (Rb=0.61 ) Deep meridional Flow (Rb=0.61 R סּ ) t=T/8 t=T/2 t=T/4 t=3T/8 Toroidal Poloidal Toroidal field Radial field

15. Exploring a prolate differential rotation Based on early helioseismic results of Charbonneau et al., 1999, we modified the r C term in eq. (6) to get a prolate shape differentail rotationr C term in eq. (6) Toroidal field Radial field

16. Differential rotation Helioseismologic observations: dΩ/dr(r=0.7 Rסּ) (6) (Schou et al., 1998)

17. Exploring a prolate differential rotation Based on early helioseismic results of Charbonneau et al., 1999, we modified the r C term in eq. (6) to get a prolate shape differentail rotationr C term in eq. (6) Toroidal field Radial field

18. CONCLUSIONS 1.If the meridional flow is confined to the convective zone, the sunspots are mainly concentrated near the poles. 2.If the flow is allowed to penetrate deeper down 0.61R, the model gives a better result since a branch of maximum toroidal field is produced at low latitudes in agreement with the observations. However, in this case another branch also appears near the poles which is inconsistent with the observed butterfly diagrams. 3.If we consider a prolate differential rotation there is a lager concentration of the toroidal field towards the equator in better agreement with the observations. 4.Both the global behavior of the circulation and the physical mechanism behind it need further revisions.