Inductive Local Correlation Tracking or, Getting from One Magnetogram to the Next

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Inductive Local Correlation Trackingor, Getting
from One Magnetogram to the Next
Brian Welsch, Bill Abbett, and George
Fisher, Space Sciences Lab, UC-Berkeley

• Goal (MURI grant) Realistically simulate coronal
  magnetic field in eruptive events.
• New Idea Evolve boundary of MHD code
  consistently with magnetograms.
• Question How can one derive a velocity from a
  time series of magnetograms?

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Outline

1. How does the field in magnetograms evolve?
2. How can we derive flows consistent with both this
   evolution and MHD?
3. How do we use these flows?

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An example of magnetic evolution in an active
region

• NOAA AR 8210, May 1 1998 1 day of evolution
  seen by MDI

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Local Correlation Tracking, Fourier-style

• Central idea of LCT schema find proper motions
  of features in a pair of successive images are by
  maximizing a cross-correlation function (or
  minimizing an error function) between sub-regions
  of the images.
• The concept is generally attributed to November
  Simon (1988).
• Useful with G-band filtergrams, H? images, or
  magnetograms.
• The FLCT method (which we developed) is similar.
  For each pixel, we
• mask each image with a Gaussian, of width ?,
  centered at that pixel
• crop the resulting images, keeping only
  signficant regions
• compute the cross-correlation function between
  the two cropped images, using standard Fast
  Fourier Transform (FFT) techniques
• use cubic-convolution interpolation to find the
  shifts in x and y that maximize the
  cross-correlation function to one of two
  precisions (chosen by the user), either 0.1 or
  0.02 pixel and
• use the shifts in x and y and ?t between images
  to find the intensity features' apparent motion
  along the solar surface.

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Example of FLCT applied to NOAA AR 8210 (May 1
1998)
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The Demoulin Berger (2003) Interpretation of LCT

• Apparent horizontal
• motion, ULCT , is from
• combination of hori-
• zontal motions and
• vertical motions acting
• on non-vertical fields.

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The Ideal MHD Induction Equation

• How can we ensure that LCT-determined velocities
  are physically consistent with the magnetic
  induction equation?
• Only the z-component of the induction equation
  contains no unobservable vertical derivatives

Now, substitute in the Demoulin Berger
hypothesis
The ideal MHD induction equation simplifies to
this form
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ILCT Use LCT to constrain solutions of the
induction equation
Let

• Solve for ?,? with 2D divergence and 2D curl
  (z-comp), and the approximation that UULCT

Note that if only Bz (or an approximation to it,
BLOS) is known, we can still solve for ?,? !
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Apply ILCT to IVM vector magnetogram data for AR
8210

• Vector magnetic field data enables us to find 3-D
  flow field from ILCT via the equations shown on
  slide 5. Transverse flows are shown as arrows,
  up/down flows shown as blue/red contours.

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Flows Consistent with Induction Equation!
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We have v(x,y,0t) --- now put it all together...
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We have v(x,y,0t) --- now put it all together...
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We have v(x,y,0t) --- now put it all together...
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We have v(x,y,0t) --- now put it all together...
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We have v(x,y,0t) --- now put it all together...
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Bill Abbett has driven an MHD code with v(x,y)!
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Conclusions

• From a time series of vector magnetograms, weve
  derived three-component, photospheric flows that
  are quantitatively consistent with the induction
  equations z-component.
• Qualitatively, the flows appear consistent with
  the observed field evolution.
• Were using these flows to drive MHD simulations
  please stay tuned!

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ILCT Flows are Only Consistent with Induction
Eqns Normal Component!
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What about other components?
Directly measured
????
Derived by ILCT
From NLFFF Extrapolation/ Prev. Time Step
at photosphere, z 0
above photosphere, z gt 0
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Incorporate Vertical Gradients with a Boundary
Code!
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What about other components?
Directly measured
Calculated by Boundary Code
Derived by ILCT
From NLFFF Extrapolation/ Prev. Time Step
at photosphere, z 0
above photosphere, z gt 0
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Induction Equations Components
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Driving Simulations from LOS Data, p.1

1. Start with a vector magnetogram.
2. Derive ?,? that will move system from vector
   data to initial LOS data.
3. Derive ?,? to move system from one LOS
   magnetogram to the next.
4. Evolve code, using B and ?,? at each step to
   find v(x,y) via

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Driving Simulations from LOS Data, p.2
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Driving Simulations from LOS Data, p.3
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Driving Simulations from LOS Data, p.4
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Driving Simulations from LOS Data, p.5
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Driving Simulations from LOS Data, p.6
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Driving Simulations from LOS Data, p.7