THE CIRCULATION DOMINATED SOLAR THE CIRCULATION DOMINATED SOLAR

DYNAMO MODEL REVISITEDDYNAMO 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

ContentsContents

• Sunspot cycle• The Solar dynamo• Mathematical Formalism

(MHD)• Results• Conclusions

SUNSPOT CYCLE FIGURESSUNSPOT 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

• ~103 G for the sunspots • Tens of G. for the diffuse poloidal field

Sunspots pair Observational butterfly diagram (NASA)

Solar cycle

THE SOLAR DYNAMOTHE SOLAR DYNAMO

1. Dipolar initial magnetic field

2. Differential rotation -> belt of toroidal field arround the solar equator.

3. Magnetic Flux tubes (ρi < ρi) -> Magnetic bouyancy.

4. Tilt bipolar active magnetic regions (BMR)

5. Decay of BMR`s.

(Fan et al. 2003)

MATHEMATICAL FORMALISMMATHEMATICAL 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

Differential rotationDifferential rotationHelioseismologic observations:

dΩ/dr(r=0.7 Rּס)

(6)

(Schou et al., 1998)

MATHEMATICAL FORMALISMMATHEMATICAL 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

Meridional CirculationMeridional CirculationA 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)

MATHEMATICAL FORMALISMMATHEMATICAL 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

Magnetic BouyancyMagnetic Bouyancy DiffusivityDiffusivity

Magnetic diffusivity for the toroidal field

Magnetic diffusivity for the poloidal field

(9) (10)

(11)

MATHEMATICAL FORMALISMMATHEMATICAL 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

TWO IMPORTANT MODELSTWO 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.

RESULTSRESULTS

Parameter Value

Uo 2000 cm/s

So 25 cm/s

ηC 2.4x1011

Rb (deep) 0.675 Rּס

We solved the equations (4) and (5) ina two dimensional mesh of 128x128spatial divisions, with 0.55<r<1 Rּס, and 0<θ<π/2 and the next boundary conditions:At θ=0 A=0 B=0At θ=π/2 dA/dt=0 B=0At r=0.55 A=0 B=0At r= Rּס (free space cond) B=0

Toroidal field

Radial field

Deep meridional Flow (Rb=0.61 Deep meridional Flow (Rb=0.61 Rּס))

t=T/8

t=T/2

t=T/4

t=3T/8

Toroidal Poloidal

Toroidal field

Radial field

Exploring a prolate differential rotationExploring a prolate differential rotation

Based on early helioseismic results of Charbonneau et al., 1999, we modified the rC term in eq. (6) to get a prolate shape differentail rotation

Toroidal field

Radial field

Exploring a prolate differential rotationExploring a prolate differential rotation

Based on early helioseismic results of Charbonneau et al., 1999, we modified the rC term in eq. (6) to get a prolate shape differentail rotation

Toroidal field

Radial field

CONCLUSIONSCONCLUSIONS

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.