H. Isobe 　 Plasma seminar 2004/06/16

1. Explaining the latitudinal distribution of sunspots with deep meridional flow D. Nandy and A.R. Choudhhuri 2002, Science, 296, 1671

2. Kinetic solar dynamo models with a deep meridional flowG.A. Guerrero and J.D. Munos 2004, MNRAS, 350, 317

3. The competition in the solar dynamo between surface and deep-seated alpha-effectsJ. Mason, D.W. Hughes, and S.M. Tobias 2002, ApJ, 580, L89

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Dynamo: origin of magnetic field

• 11(22) year cycle• Preferred longitude of sunspot emergence• Twist (helicity) as an origin of surface activity

Kinematic/Dynamic Dynamo

MHD induction eq.

• If plasma velocity is given, the induction equation is linear and the problem is called kinematic (linear) dynamo.• When back reaction of the B on U is considered, one has to solve momentum equation and hence the problem is nonlinear. It is called dynamic (nonlinear) dynamo.

αωdynamo

Generate toroidal field from poloidal field by stretching the field line by differential rotation.

Generate poloidal field from toroidal field by Colioris force, turbulence (concection), MHD instability etc.

Explaining the Latitudinal Distribution of Sunspots with Deep Meridional Flow

D. Nandy & A.R. Choudhuri 2002, Science, 296, 1671

• Kinematic dynamo model using rotational velocity profile below the surface obtained by helioseismology and meridional flow.

• By considering meridional flow penetrating the tachocline, they successfully explain the latitudal distribution of the sunspots.

Mathematical formulation (1)

↑toroidal 　↑ poloidal

↓αeffect

• 2D axisymmetric induction equation with αeffect.• They assume that αeffect works only near the surface.

Mathematical formulation (2)

From helioseismology Model in the simulation

•Strong velosity shear at the base of the convection zone (tachocline) => ωeffect

•Buoyancy algorism: if B>105G, halr of the flux is made to erupt to the surface layers.

•They consider meridional flows (1) only in the convection zone and (2) penetrating below the tachocline into the radiative zone,

Meridional circulation flow (observationaly unknown)

Eruption latitude vs time plot of sunspots

Meridional flow only in the convection zone

Meridional flow penetrating below the tachocline

Sunspots appear in high latitude region (inconsistent with observation) if the Meridional flow does not penetrate into the stable (radiateve) zone.

Kinematic dynamo scenarionegative flux

positive flux

•ωeffect is effective in high latitude tachocline because of strong shear.•Magnetic flux is stored in the stable (radiative) zone by penetrating flow•Transport of flux to lower latitude by Meridional circulation•Erupt to surface (low latitude) by buoyancy, formation and decay of active region (αeffect)•Transport of flux to lower latitude and in the convection zone by Meridional curculation

Kinematic Solar Dynamo Models with a Deep Meridional Flow

• Similar kinematic model to that of Nandy and Choudhuri (2002)

• Different treatment of αeffect, buoyancy, and density profile

• The results show some difference from that of Nandy and Choudhuri. In particular, the result using more realistic density profile differ from observation.

• However, the role of the deep penetrating Meridional flow seems to be robust.

Mathematical formulation (different points from Nandy and Choudhuri model)

1.αeffect and buoyancy (Dikpati & Charbonneau 1999)

Mathematical formulation (different points from Nandy and Choudhuri model)

2. Density profile

For density profile they use:(1) Adiabatic stratification with si

ngle polytrope (γ=5/3)(2) Adiabatic straticfication with γ

=5/3 in the convection zone and γ=1.26 in the radiative zone.

Result 1. with singpe polytropic density profileMeridional flow only in CZ Meridional flow in deeper zone

toroidal B at r=0.7Ro

poloidal B at r=Ro

Basic tendency is consistent with Nandy and Choudhuri.

But there is a high latitide peak in the deep Meridional flow case.

Result 2. with bipolytropic density profile

Peak at high latitude.Period is also longer (72.2 yr) than previous case (28.8 yr)

Conclusion

• If the Meridional flow is confined in the convection zone, the emergence latitude of sunspots are higher than observation.

• If the Meridional flow penetrate into the stable zone, the emergence latitude is lower. But they also find a high latitude peak, inconsistent with Nandy and Choudhuri

• Using more realistic density profiles results in worse results, e.g., longer period.

• Dynamics of buoyant breakup and rise of the flux tube should be studied. => 3D MHD simulation.

The Competition in the Solar Dynamo between Surface and Deep-Seated α-effect

J. Mason, D.W. Hughes, and S.M. Tobias 2002, ApJ, 580, L89

• Examine efficient location for α-effect. Near the surface .vs. near the base.

• Linear analysis of induction equations (advection-duffusion equations)

• They found that α-effect near the base of the convection zone is more effective.

Background• It is established that ω-effect operates at the

base of the convection zone (tachocline)• Location of α-effect is an open question

– Classical view by Parker (1955) -- α-effect by cyclonic convection, hence throughout the convection zone

– In Badcock & Leighton model, the poloidal field is produced by the decay of active region (α-effect near the surface)

– α-effect near the base of the convection zone by e.g., instability of the magnetic layer.

• Reason why some authors believe α-effect near the surface (e.g. Nandy, Choudhuri)– Simulation of rise of flux tube by thin flux tube mod

el predict that the toroidal field at the base of the convection zone must be B>105 G

– 105 G is an order of magnetitude larger than the equipartition valule of the convecitive flows, therefore convection cannot bent the field line, i.e., α-effect cannot operate.

• Argument of Hughes-san and authors of this paper– “I don’t believe the thin flux tube calculations at all”– It seems more natural that the locations of α-effect

and ω-effect are close.– Instability of the magnetic layer (e.g. Parker instabili

ty) can also generate the poloidal field.

Mathematical formulation (1)(i) Basic equations

(ii) ω-effect at z=0 (base of convection zone)

(iii)α-effect at z=1 (surface) and z=λ<1

(iv) Parameters are D (dynamo number),ε(ratio of two competing α-effects, and λ(location of the second α-effect)

-L 0 λ 　　 　 1 L

ω 　 αi 　　　 αs

Mathematical formulation (2)

* No Meridional flow is considered.

(v) Matching conditions and boundary conditions.

(vi) Seek solutions of the form:

(vii) Obtain dispersion relation (q=p+k2)

Result(1)

q=p+k2 ,p=σ+iω

•For given wavenumber k, Dc denote the dynamo number D at which growth rate σ=0, and frequency at this point ωc. •In the case of ε=0 (only surface α-effect), they found, that the system support both positive and negative frequency, incontrast earliear studies assuming unform α-and ω-effects (Parker 1995, Plasma Astrophysics 3.1.1.5).z

x

<= In this case, positive frequency = southward propagationnegative frequency = northward propagation

Result (2) k-Dc diagram

solid lines: ε=0 (only surface α-effect)dashed lins: ε=0.01, λ=0.1