H. Isobe Plasma seminar 2004/06/16

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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 flow G.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

• H. Isobe Plasma seminar 2004/06/16

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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

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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.

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a?dynamo
Generate poloidal field from toroidal field by
Colioris force, turbulence (concection), MHD
instability etc.
Generate toroidal field from poloidal field by
stretching the field line by differential
rotation.
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Explaining the Latitudinal Distribution of
Sunspots with Deep Meridional FlowD. 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.

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Mathematical formulation (1)
?toroidal ?poloidal
?aeffect

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

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Mathematical formulation (2)
From helioseismology
Model in the simulation

• Strong velosity shear at the base of the
  convection zone (tachocline) gt ?effect
• Buoyancy algorism if Bgt105G, halr of the flux is
  made to erupt to the surface layers.

Meridional circulation flow (observationaly
unknown)

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

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Eruption latitude vs time plot of sunspots
Meridional flow penetrating below the tachocline
Meridional flow only in the convection zone
Sunspots appear in high latitude region
(inconsistent with observation) if the Meridional
flow does not penetrate into the stable
(radiateve) zone.
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Kinematic dynamo scenario
negative flux
positive flux

• ?effect 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 (aeffect)
• Transport of flux to lower latitude and in the
  convection zone by Meridional curculation

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Kinematic Solar Dynamo Models with a Deep
Meridional Flow

• Similar kinematic model to that of Nandy and
  Choudhuri (2002)
• Different treatment of aeffect, 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.

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Mathematical formulation (different points from
Nandy and Choudhuri model)
1.aeffect and buoyancy (Dikpati Charbonneau
1999)
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Mathematical formulation (different points from
Nandy and Choudhuri model)
2. Density profile

• For density profile they use
• Adiabatic stratification with single polytrope
  (?5/3)
• (2) Adiabatic straticfication with ?5/3 in the
  convection zone and ?1.26 in the radiative zone.

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Result 1. with singpe polytropic density profile
Meridional flow only in CZ Meridional flow in
deeper zone
toroidal B at r0.7Ro
poloidal B at rRo
Basic tendency is consistent with Nandy and
Choudhuri. But there is a high latitide peak in
the deep Meridional flow case.
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Result 2. with bipolytropic density profile
Peak at high latitude. Period is also longer
(72.2 yr) than previous case (28.8 yr)
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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. gt 3D MHD simulation.

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The Competition in the Solar Dynamo between
Surface and Deep-Seated a-effectJ. Mason, D.W.
Hughes, and S.M. Tobias 2002, ApJ, 580, L89

• Examine efficient location for a-effect. Near the
  surface .vs. near the base.
• Linear analysis of induction equations
  (advection-duffusion equations)
• They found that a-effect near the base of the
  convection zone is more effective.

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Background

• It is established that ?-effect operates at the
  base of the convection zone (tachocline)
• Location of a-effect is an open question
• Classical view by Parker (1955) -- a-effect by
  cyclonic convection, hence throughout the
  convection zone
• In Badcock Leighton model, the poloidal field
  is produced by the decay of active region
  (a-effect near the surface)
• a-effect near the base of the convection zone by
  e.g., instability of the magnetic layer.

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• Reason why some authors believe a-effect near the
  surface (e.g. Nandy, Choudhuri)
• Simulation of rise of flux tube by thin flux tube
  model predict that the toroidal field at the base
  of the convection zone must be Bgt105 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., a-effect cannot operate.
• Argument of Hughes-san and authors of this paper
• I dont believe the thin flux tube calculations
  at all
• It seems more natural that the locations of
  a-effect and ?-effect are close.
• Instability of the magnetic layer (e.g. Parker
  instability) can also generate the poloidal
  field.

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Mathematical formulation (1)
(i) Basic equations
(ii) ?-effect at z0 (base of convection zone)
(iii)a-effect at z1 (surface) and z?lt1
(iv) Parameters are D (dynamo number),e(ratio of
two competing a-effects, and ?(location of the
second a-effect)
? ai as
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Mathematical formulation (2)
(v) Matching conditions and boundary conditions.
(vi) Seek solutions of the form
(vii) Obtain dispersion relation (qpk2)
No Meridional flow is considered.
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Result(1)
qpk2 ,psi?

• For given wavenumber k, Dc denote the dynamo
  number D at which growth rate s0, and frequency
  at this point ?c.
• In the case of e0 (only surface a-effect), they
  found, that the system support both positive and
  negative frequency, incontrast earliear studies
  assuming unform a-and ?-effects (Parker 1995,
  Plasma Astrophysics 3.1.1.5).

z x
lt In this case, positive frequency southward
propagation negative frequency northward
propagation
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Result (2) k-Dc diagram
solid lines e0 (only surface a-effect) dashed
lins e0.01, ?0.1
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