Nonlinear Evolution of the Magnetic Buoyancy Instability

K. Nagashima, P. Antolin, H. Miyahara, M. I. Hakim,

K. Takahashi, D. Shiota, and S. Nozawa

Emerging Flux

Magnetic flux emerges from the solar interior into the atmosphere. In the Chromosphere this emerging flux can be observed like arch filament features.

From Chou & Zirin 1988

Size ～ 30,000 km

Width ～ 20,000km

Life time ～ 3 days

Rise Velocity ～ 10-20 km/s

30,000 km

30,000 km

Parker Instability

Following the perturbation, dense material above the magnetic flux is driven along the magnetic field lines, creating a net upward force.

Shibata et al. (1989)

Result of 3D MHD simulation

H

Since the scale height of the photosphere is smaller than the thickness of a magnetic sheet, the gas pressure outside the sheet decreases rapidly.

H

The magnetic flux expands horizontally until the magnetic pressure of the sheet is balanced with the surrounding gas pressure.

Nozawa (2005)

Result of 2D MHD simulation

Vertical expansion speed and Alfven speed are found to be independent of time, being only dependent on height above Photosphere. Their ratio is found to be constant.

It is of interest to see if this result remains true for a 2.5D MHD simulation, and for different angles of magnetic field direction.

What determines the horizontal expansion (Vx) is unclear. We are interested in trying to explain this result.

Vx=?

Initial condition

Plasma beta = 1 (=p_gas/p_mag)

2.5-D ideal MHD

Uniform gravity

The modified Lax-Wendroff scheme

The simulation code CANS

Grid : 108 (x) * 158 (y)

Initial condition

t=0.0

Color :

Initial magnetic field

Normalization

Gravitational acceleration

Number density

Temperature

Normalized by photospheric values

Length unit : Photospheric scale height H ~ 200km

Velocity unit : Photospheric sound speed Cs ~10km/s

Time unit : 20sec

Result

Movie (IDL)

Density profile in a typical case

t=0.0

t=10.0

t=20.0

40H

25H

t=30.0

t=40.0

Parker Instability case

Density Profile for different values of theta

0 degree

30 degree

45 degree

80 degree

Height [H]

25

0

0

50

t [20sec]

0.49Cs

Height [H]

25

0

theta=30

0.5Cs

Height [H]

25

0

theta=45

0.43Cs

theta=0

theta=80

Height [H]

25

0

0

0

t [20sec]

t [20sec]

t [20sec]

0.22Cs

Height [H]

25

0

theta=30

Height [H]

25

0

theta=45

theta=0

theta=80

Height [H]

25

0

Height-Time profile of magnetic field (at center of loop)

Height [H]

25

0

0

50

t [20sec]

0.49Cs

0.5Cs

0.43Cs

0

0

t [20sec]

t [20sec]

t [20sec]

0.22Cs

Horizontal expansion at a height of 10H

0

25

0

25

0

25

x [H]

0

25

x [H]

x [H]

x [H]

t [20sec]

t [20sec]

t [20sec]

t [20sec]

theta=30

theta=0

theta=45

theta=80

Results

Velocity : 1 =Cs(photospheric sound speed) = 10km/s

Height : 1 = H (photospheric scale height) = 200km

(** : same loop at different times in its expansion at the loop head)

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<

<

<

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>

Velocity variation with angle

Vx

* Vy

Va

Lines:

V_0*cos(theta)

( V_0 = V(theta=0) )

Future work

Force balance in the corona

2006-03-16T16:48:01