Lorentz force: a possible driving force for sunspot rotation

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Lorentz force a possible driving force for
sunspot rotation

• J.T. Su1, Y. Liu2, J.H. Liu3, X.J. Mao4,
• H.Q. Zhang1, H. Li5, X.F. Wang1 and W.B. Xie1
• 1. National Astronomical
  Observatories, Chinese Academy of Sciences,
  Beijing 100012,China
• 2. Institute for Astronomy,
  University of Hawaii, 2680 Woodlawn Drive,
  Honolulu, HI, 96822, USA
• 3. Department of Physics, Shi
  Jiazhuang University, Shi Jiazhuang 050081, China
• 4. Astronomy Department,
  Beijing Normal University, Beijing 100875, China
• 5. Purple Mountain
  Observatory, Chinese Academy of Sciences, Nanjing
  210008, China

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Observations two opposite sub-photospheric
vortical flows

• (1) Zhao and Kosovichev (2003) found two
  opposite sub-photospheric vortical flows in the
  depth range of 0-12 Mm around a fast rotating
  sunspot.
• (2) As a result, they believed that the flux
  tubes would become more twisted and the magnetic
  enery much increase.

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Background image showing the vertical velocities
and arrows showing the horizontal velocity field

Zhao and
Kosovichev (2003)
NOAA 9114
0-3(5) Mm
2000-8-7
2000-8-8
9-12 Mm
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Main characteristics of two vortical flows
Zhao and Kosovichev (2003)
In addition, by checking the vector magnetograms
of the sunspot on 9 August 2000, we found that
the longitudinal field was positive polarity and
the transverse field exhibited a CW trend.
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Motivation

• Sunspot that exhibits some degrees of rotational
  motion around its own vertical
• axis is not rare in solar observations.
  Generally, sunspot rotation is considered
• as resource of the magnetic energy (or helicity)
  accumulation.
• Is it ture that sunspot rotation can build up the
  magnetic energy (or helicity)?
• Is Lorentz force (LF) a possible mechanism for
  driving sunspot rotation?
• Thus, we want to make a qualitative analysis of
  the LF on a sunspot.

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A qualitative analysis of the LF on a sunspot

• The equation of motion for an incompressible
  flow

We try to get the integral formula of for a
sunspot model under the photosphere.
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The LF on sunspot under the photosphere

• There are three integral surfaces
• of the sunspot needed to be
• considered under the photosphere
• lateral surfaces of flux tubes
• photosphere
• (c) a certain bottom surface.
• Note here we assume there is a certain bottom at
  which the flux axis is not deformed and it could
  not be the convection zone.

Parker (1979)
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In Cartesian
coordinates Lateral surface integral Top
surface integral (Photosphere z0) Bottom
surface integral
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• If the magnetic flux tubes of sunspot are
  twisted, then the magnetic torque
• acting on a segment of the tube is (Longcope and
  Klapper, 1997)

• bottom
  surface top surface
• There were two possibilities for the relationship
  between the opposite vortical
• flows and the twist of a flux tube
• two opposite vortical flows may creat the twist
  of a flux tube
• (2) the twisted flux tube may creat two opposite
  vortical flows.

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Two opposite vortical flows may create the twist
of a flux tube
(seen from the top)

• A CCW twist is introduced in the flux tube,
  however, which disagrees with the 2000 August 9
  observations that the transverse field exhibited
  a CW trend.
• (2) The flux tube must contract with the twist
  increasing and the contraction would lead to
  upward flows at the upper region (e.g., 0-3(5)
  Mm), and downward flows at the lower region
  (e.g., 9-12) Mm.

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• The twist of a CW flux tube must decrease as a
• CCW force on the top or a CW force on the bottom

init_twist
after_CCW_rotation
Provided by the anonymous referee
Two driving forces may origin from the twist of a
flux tube.
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A already twisted flux tube may create two
vortical flows

• A flux tube with CW twist ( ) is maintained
  by a CCW magnetic
• force (F1) on the bottom and a CW magnetic
  force (F4) on the top of it.
• (2) Correspondingly, there would be two
  counterforces CF1 (CW) and
• CF4 (CCW) acting on the surrounding plasma
  of the flux tube.
• (3) The CF1 may create CW vortical flow at the
  bottom and the CF4
• may create CCW vortical flow at the top
  when a flux tube emerges
• cross the photosphere.
• (4) Thus, the twist of the flux tube would
  decrease and the magnetic
• pressure in the tube will decrease, which
  will cause gas to flow into
• the tube from above and below to establish
  pressure equilibrium.

A flux tube with CW twist
This is actually the observations of Zhao and
Kosovichev (2003) a downflow into the tube from
above, and an upflow into the tube from below.
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Another case of fast rotating sunspot on 11
December 2006
The angular acceleration of the rotation was
-5.010-8 deg s-2 (CW is minus). Similarly, Brown
et al. (2003) observed rotating sunspots
accelerating from rest to speeds of 1 deg hr-1
over 20 hours, or about 410-9 deg s-2.
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Fy3
Fy4
Minor sunspot
In Box 3 Fy3-3.281021 N, In Box 4
Fy42.731021 N, On the photosphere
Ttop5.81027 N m. Some parameters of sunspot in
the depth range of,e.g. 0-5 Mm L5 Mm,
?3.0102 kg m-3, M7.51022 kg and r4
Mm Angular acceleration a 5.0 10-7 deg s-2
(CCW).
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A constrain on the field line twist at a bottom
of flux tube

• That the observed angular acceleration -5.110-8
  deg s-2 (CW) is less than the calculated
• one 5.0 10-7 deg s-2 means that we must include
  the magnetic torque contribution from
• a certain bottom surface (e.g., depth 5Mm) and
  the bottom torque was greater than the top
• torque.

If two magnetic field compoents satisfying the
constraint BFqrBz are assumed, where q is the
tubes pitch at the depth of 5 Mm, we can get the
value of q 6.2510-3 m-1.
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Non-potential magnetic stress
The dimension of magnetic stress is the force per
unit area or the energy per unit volume
We define such a vector, the non-potential
magnetic stress (NPMS) as follows
where the second terms are the corresponding
potential quantities extrapolated from the
observed longitudinal magnetic field.
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Time sequence of non-potential mangetic stress

• The LOS component is presented
• by the gray-scale patches and the
• transverse component by the arrows.
• The magnetic field observations were
• taken from Huairou full-disk vector
• Magnetograph and Hinode SP/SOT.
• The increase of LOS componet
• along the MNL was obvious from
• 0046 UT to 0302 UT (the start of
• flare X3.4 at 0228 UT).
• (2) The transverse stress first points
• toward the MNL, then along it, and
• finally it points away from it.
• We need other data to verify whether or

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• Summary
• We think that sunspot rotation is not the process
  of building up the magnetic energy or twist.
  However, it may just represent the inverse
  process.
• The photosphere motion and subphotospheric
  vortical flows of sunspot may be driven by
  Lorentz force.
• We provide a vector quantity, non-potential
  magnetic stress to study flare evolution.

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• Thanks for your attention!