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Formation and Eruption ofFilament Flux Ropes

A. A. van Ballegooijen1 & D. H. Mackay2

1 Smithsonian Astrophysical Observatory, Cambridge, MA2 University of St. Andrews, Scotland, UK

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Filament Flux RopesFilaments are cool plasma (~104 K) embedded in hot corona and supported by a magnetic flux rope:

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Formation of Flux Ropes by Reconnection

Pneuman (Sol. Phys. 88, 219, 1983) proposed that helical flux ropesare formed by reconnection in the low corona:

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Formation of Flux Ropes by Reconnection

Van Ballegooijen & Martens (ApJ, 343, 971, 1989) proposed reconnection is associated with magnetic flux cancellation in a sheared arcade:

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Modeling Coronal Magnetic Fields

.BBvA×∇−×=

∂∂ ηt

Mean-field approach to modeling the evolution of coronal magnetic fields(van Ballegooijen et al., ApJ, 539, 983, 2000):

• The coronal magnetic field is sum of mean and fluctuating components.• The mean field B(r,t) describes the large-scale structure of the corona.• The fluctuating field δB(r,t) describes small-scale twists of the coronal field lines, produced by footpoint motions on the scale of the solar granulation ( ~ 1 Mm).• Small-scale reconnection events in the corona cause diffusion of the mean magnetic field. Such events are responsible for coronal heating (e.g., Parker 1972).• Simplest possible model: isotropic diffusion.

AB ×∇=

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Modeling Coronal Magnetic FieldsAt the photosphere, the mean field is subject to large-scale surface flows (vθ and vφ) and diffusion (D):

,,sin θφθ φ

φφ

θ

∂∂

+−=∂∂

∂∂

−=∂∂ r

rr

rB

rDBv

tAB

rDBv

tA

Surface diffusion allows for the observed flux cancellation at polarity inversion lines.

Above model assumes there is only horizontal diffusion of the radial field, no radial diffusion of horizontal field. Therefore, once formed, coronal flux ropes cannot “submerge” below the photosphere.

Observations indicate D ≈ 600 km2 s-1 (Wang et al. 1989).For coronal diffusion driven by photospheric footpoint motions, η ~ D.

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Evolution of Two Sheared BipolesMackay & van Ballegooijen (ApJ, 641, 577, 2006) simulated the interaction of two bipolar active regions using a mean-field approach. Both bipoles have sheared magnetic field along the polarity inversion line (PIL):

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Evolution of Two Sheared BipolesStrong magnetic shear develops on the internal PIL of the trailing bipole on day 18 (left), and on the external PIL on day 40 (right):

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Evolution of Two Sheared Bipoles

Lift-off of the flux rope lying above the internal PIL of the trailing bipole:

day 17 day 22 day 25

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Evolution of Two Sheared BipolesReconnection of field lines below the erupting flux rope:

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Evolution of Two Sheared BipolesHeights of flux ropes (solid lines) and magnetic nulls (dashed) vs. day of simulation (C=initial null, T=trailing bipole, L=lead bipole, E=external):

DAY OF SIMULATION

RA

DIU

S (s

olar

radi

i)

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Evolution of Two Sheared BipolesMackay & van Ballegooijen (ApJ, 642, 1193, 2006) constructed connectivity maps (blue=bipolar, green=cross-bipolar, red=open, yellow=periodic):

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Improved Models of Diffusion

( ) ( )

,

,

2

2422

B

BBt

BB

BBBvA

×∇⋅≡

∇⋅∇+×∇−×=∂∂

⊥

λ

ληη

What is the effect of coronal reconnection on magnetic helicity?

where η2 describes perpendicular diffusion, and η4 describes hyper-diffusion (Boozer 1986; Strauss 1988).

If reconnection (and heating) events are highly localized, H ≈ constant.To satisfy this helicity constraint, the mean-field induction equation can be modified as follows:

∫ ⋅= dVH BA

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Evolution of an Ω-loopSimulate coupled evolution of an Ω-loop with perpendicular diffusion in the convection zone (η2 = 600 km2 s-1) and hyper-diffusion in the corona(η4 = 3×1011 km4 s-1). Initial configuration produced by pushing a twisted flux rope up from the base of the convection zone:

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Evolution of an Ω-loopStrong magnetic shear builds up along the polarity inversion line:

t = 0 days t = 4 days

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Evolution of an Ω-loopAfter 10 days, sufficient flux has cancelled at photosphere for the coronal field to erupt:

Contours of Bz in ahorizontal slice (z=0.1 R):

Contours of λ in a vertical slice:

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Evolution of an Ω-loopGlobal configuration of the magnetic field after 10 days:

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Conclusions• Filaments/prominences are embedded in highly sheared, weakly twisted fields (“coronal flux ropes”).• These coronal flux ropes are not identical to the Ω-loops that emerge through the photosphere. The emerged field is reconfigured by reconnection processes that enhance the magnetic shear.• Two types of reconnection are important: a) reconnection associated with photospheric flux cancellation; b) small-scale reconnection which produces “diffusion” of the mean coronal magnetic field and reduces the degree of twist of flux ropes.

The mean field approach is an effective tool for modeling the evolution of coronal and sub-surface magnetic fields over long periods (many days). Magnetic helicity constraints have been included into such models.