Modeling the Magnetic Field Evolution of the December 13 2006 Eruptive Flare

Yuhong FanHigh Altitude Observatory, National Center for Atmospheric Research

FEW 2011

Outline• A set of simulations of CME onset with an idealized

configuration:− Consider a pre-existing coronal potential arcade field and impose the

emergence of a twisted flux rope at the lower boundary− Critical conditions for the eruption of a coronal flux rope− Formation of current sheet and the role of “tether cutting”

reconnections• An observationally guided simulation

− Both the pre-existing field and the lower boundary driving conditions are derived to some degree from observations

− Qualitatively models the magnetic field evolution associated with the December 13 2006 eruptive flare

MHD simulations of the eruption of coronal flux ropes

• Numerically solve the isothermal MHD equations in a spherical domain of the solar corona:

• The domain is resolved by a non-uniform grid of 432x192x240

• Initially the corona is a static isothermal atmosphere at 1MK with a pre-existing potential arcade field: the isothermal sound speed as=128km/s, the peak Alfven speed at the foot point of the arcade vA0=1951km/s.

• At the lower boundary, we impose (kinematically) the emergence of a twisted torus for t=0 to t=tstp after which the emergence is stopped and the field lines are rigidly anchored subsequently.

• A sequence of simulations are carried out where tstp is varied such that a varying amount of the twisted flux of the torus is transported into the corona.

€

r∈ Rs, 5.496Rs[ ], θ ∈ 5π /12, 7π /12[ ], φ∈ −π /9.6, π /9.6[ ]

Fan (2010)

When does dynamic eruption occur?

€

tstp = 89 Rs /vA0, Hm = 0.20205Φ2

€

tstp = 89.125 Rs /vA0, Hm = 0.20215Φ2

€

At the critical height : - d lnBp /dh =1.74

Orange surfaces: iso-surfaces of J/B with the level set at 1/l where l = 10 grid resolution elements.

Formation of current sheet and “tether cutting” reconnections

Hinode observation of the d-region NOAA 10930 and the eruptive flare on 2006-12-13

Images and movies from http://solar-b.nao.ac.jp/news_e/20061213_flare_e.shtml

Min and Chae (2009)

• The small sunspot of positive polarity rotated counter-clockwise about its center by 240° as measured by Zhang et al. (2007) and 540° as measured by Min and Chae (2009).

Liu et al. (2008)

Hinode observation of the d-region NOAA 10930 and the eruptive flare on 2006-12-13

Images and movies from http://solar-b.nao.ac.jp/news_e/20061213_flare_e.

shtml

• Constructing the initial pre-existing field and the lower boundary driving conditions

•A region centered on the d-spot is extracted from the MDI full disk magnetogram

• Smoothing of Br with a Gaussian filter• The magnetic flux in a central area enclosing the region

of flux emergence is zeroed out• Construct potential field from the lower boundary

normal flux distribution as the pre-existing coronal field• On the lower boundary, in the zeroed out area, drive the

emergence of an idealized, twisted magnetic torus.

20:51:01 UT on Dec. 12, 2006

initial normal flux distribution final normal flux distribution

• We solve the following MHD equations, assuming an ideal polytropic gas with g = 1.1:

Simulation domain:

Grid:

€

512 × 352 × 528

Initial atmosphere is assumed to be a static polytropic atmosphere with g=1.1.

Initial potential magnetic field

3D coronal magnetic field evolution

t = 3.25

t = 3.55

€

where vA 0 =1951km/s

t = 3.65

Liu et al. (2008)

€

Orange surface :isosurface ofJ /B =1/(10 × dr)and where ΔS /CV >1.15

€

t = 2.45

€

t = 2.45

€

t = 2.45

€

Orange surface :isosurface ofJ /B =1/(5 × dr)and where ΔS /CV > 2.3

Evolution of post-flare loops

€

t = 2.8

€

t = 3.15

€

t = 2.8

€

t = 3.15

Evolution of flare ribbons

€

t = 2.8

€

t = 3.15

Summary• The simulated coronal magnetic field resulting from the emergence of an east-west

oriented flux rope with its positive emerging flux bordering the southern edge of the dominant pre-existing negative sunspot captures the gross structure of the actual magnetic field evolution associated with the eruptive flare (Fan 2011 ApJ in press).

• Improvement of the model:o Much wider simulation domain

− Increase spatial decline rate of the ambient potential field faster eruption

− Remove the interference of the sidewall boundaries on the trajectory and writhing of the erupting flux rope

o Reduce smoothing of the observed lower boundary flux densityo More quantitative determination of the lower boundary electric field that

results in better matching of the observe flux emergence pattern.

AcknowledgementsThis work is supported in part by NASA LWS TR&T grant NNX09AJ89G to NCAR. The numerical simulations were carried out on the Pleiades supercomputer at the NASA Advanced Supercomputing Division.