3D Spherical Shell Simulations of Rising Flux Tubes in the Solar Convective Envelope

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3D Spherical Shell Simulations of Rising Flux
Tubes in the Solar Convective Envelope

• Yuhong Fan (HAO/NCAR)

High Altitude Observatory (HAO) National Center
for Atmospheric Research (NCAR) The National
Center for Atmospheric Research is operated by
the University Corporation for Atmospheric
Research under sponsorship of the National
Science Foundation. An Equal Opportunity/Affirmati
ve Action Employer.
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Outline

• Overview of results from the thin flux tube model
  and from MHD simulations in local Cartesian
  geometries
• New results from simulations of buoyantly rising
  magnetic flux tubes in the solar convective
  envelope using a spherical shell anelastic MHD
  code.

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Full disk magnetogram from KPNO
Figure by George Fisher
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The Thin Flux Tube Model

• Thin flux tube approximation
• all physical quantities are
  averages over the tube cross-section, solve for
  the mean motion of each tube segment under the
  relevant forces
• Results
• Field strength of the toroidal magnetic field at
  the base of SCZ is of order
• Tilt of the emerging loop active region tilts,
  Joys law
• Asymmetric inclination of the two sides of the
  emerging loop
• Asymmetric field strength between the two sides
  of the loop

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MHD Simulations in Local Cartesian Geometries

• The dynamic effects of field-line twist

• Maintaining cohesion of rising flux tubes

Abbett et al. (2001)
Fan et al. (1998)
untwisted
untwisted
twisted
twisted
For 2D horizontal tubes twist rate
, where (e.g.
Moreno-Insertis Emonet 1996, Fan et al. 1998
Longcope et al. 1999). For 3D arched flux tubes
necessary twist may be less, depending on the
initial conditions (e.g. Abbett et al. 2000 Fan
2001)
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Apex cross-section
Fan (2001)
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• The dynamic effects of field-line twist
  (continued)

• Becoming kink unstable when the twist is
  sufficiently high ? formation of flare-productive
  delta-sunspot regions (e.g. Linton et al. 1998,
  1999 Fan et al. 1999)

Fan et al. (1999)
twist rate , where
(Linton et al. 1996)
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Anelastic MHD Simulations in a Spherical Shell
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Anelastic MHD Simulations in a Spherical Shell

• We solve the above anelastic MHD equations in a
  spherical shell representing the solar convective
  envelope (which may include a sub-adiabatically
  stratified stable thin overshoot layer)
• staggered finite-difference
• two-step predictor-corrector time stepping
• An upwind, monotonicity-preserving interpolation
  scheme is used for evaluating the fluxes of the
  advection terms in the momentum equations
• A method of characteristics that is upwind in
  the Alfven waves is used for evaluating the V x B
  term in the induction equation (Stone Norman
  1992).
• The constrained transport scheme is used for
  advancing the induction equation to ensure that B
  remains divergence free.
• Solving the elliptic equation for at every
  sub-time step to ensure
• FFT in the -direction ? a 2D linear system
  for each azimuthal order
• The 2D linear equation (in ) for each
  azimuthal order is solved with the
  generalized cyclic reduction scheme of
  Swartztrauber (NCARs FISHPACK).

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Anelastic MHD Simulations in a Spherical Shell
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central cross-section of -tube
axisymmetric
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• In the axisymmetric case, the angular momentum of
  each tube segment is conserved.

axisymmetric
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central cross-section of -tube
-tube
axisymmetic
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• A twisted flux tube when arched upward will
  rotate out of the plane, i.e. develop a writhe.
• For a left-hand-twisted (right-hand twisted)
  tube, the rotation is counter-clockwise
  (clockwise) when viewed from the top.

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Apex cross section
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For the emergence of a left-hand-twisted flux
tube, the polarity orientation starts out as
south-north oriented, and then after an apparent
shearing motion, establishes the correct tilt.
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Summary

• From the simulations of the buoyant rise of
  -shaped flux tubes in a rotating spherical model
  solar convective envelope, it is found
• The rise trajectory for a 3D -tube is more
  radial than that of an axisymmetric toroidal flux
  ring.
• A twisted flux tube when arched upward develops
  a tilt that is counter-clockwise (clockwise) when
  viewed from the top if the twist is left-handed
  (right-heanded). Since flux tubes in the
  northern hemisphere are preferentially left-hand
  twisted, the twist is driving a tilt opposite to
  the effect of the Coriolis force and opposite to
  the direction of the observed mean active region
  tilt. We find that in order for the buoyant flux
  tube to emerge with a tilt consistent with
  observations, the twist of the flux tube needs to
  be less than half of the critical twist necessary
  for the tube to rise cohesively. Under such
  conditions, severe flux loss ( gt 50 of the total
  flux) is expected during the rise.

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Summary (cont.)

• Due to the asymmetric stretching of the rising
  -tube by the Coriolis force, a field strength
  asymmetry develops with the leading side of the
  emerging tube being greater in field strength and
  more cohesive compared to the following side.
  This provides a natural explanation of the
  observe morphological asymmetry of solar active
  regions where the leading polarity of an active
  region tends to be more cohesive, usually in the
  form of a large sunspot, while the following
  polarity tends to appear more fragmented.
• A retrograde flow of about 100m/s is present in
  the apex segment of the rising -tube. This
  may be a deep signature to look for to detect
  rising active region flux tubes prior to their
  emergence?

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Future work

• Self-consistently model the formation and rise
  of buoyant flux tubes from the base of the solar
  convection zone
• What are the instabilities that can lead to the
  formation of active region scale flux tubes, e.g.
  magnetic buoyancy instabilities (modified by
  solar rotation)?
• What determines the twist of the magnetic flux
  tubes that form, given the current helicity of
  the magnetic fields generated by the dynamo?
• Incorporating convection into the simulations
• Is convection important in determining the
  properties of emerging active region flux tubes?