the circulation dominated solar dynamo model revisited gustavo a. guerrero e. (iag/usp) elisabete m....
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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.
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