t. magara- 3-dimensional evolution of a magnetic flux tube emerging into the solar atmosphere

8/3/2019 T. Magara- 3-dimensional Evolution of a Magnetic Flux Tube Emerging into the Solar Atmosphere

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NATO Advanced Research Workshop

3-dimensional Evolution of a Magnetic Flux TubeEmerging into the Solar Atmosphere

T. Magara(Montana State University, USA)

September 17, 2002 (Budapest, Hungary)


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Corona :

Because the background gas pressure is weaker than mag-netic pressure,

the magnetic field continues to expand outward.

(magnetic-field dominant region)

A well-developed magnetic structure is formed.This structure is macroscopically static (force-free), however

outermost area... dynamic (solar wind) prominence area... mass motion along B

coronal loops... mass motion along B

Sometimes explosive events (relaxation of magnetic energy) happen insuch a well-developed structure. These events are

flares (produce high energy particles & electromagnetic waves) prominence eruptions (cool material erupts and disappears) coronal mass ejections (large amount of mass is ejected into IP)


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To study various coronal processes, we initially assume a skeleton of magnetic

field which provides a model of well-developed coronal structure. Then we in-vestigate its stability and evolution at both linear and nonlinear phases.

Initial stage:set a skeleton of magneticfield in the atmosphere(eg. potential, force-free, orany magnetohydrostatic

states)

Input stage:

impose various typeof perturbations to thesystem according tothe primary purposeof studies

perturbation

relaxation state:

To see the thermal and

dynamical evolutions atthe linear and nonlinearphases

magnetic reconnection


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Convection zone:

Gas dominant region

>amplification of magnetic field by plasma motion

Galloway and Weiss (1981)

The convection zone is devided into two regions:

flux tube region &field-free region

The dynamics of magnetic field in the convection zone is described by

... thin flux tube model (Defouw 1976; Roberts & Webb 1978; Parker 1979; Spruit 1981)

Various works based on thin flux tube model have provided an important knowledge of

Storage of magnetic field at the base of the convection zone

Stability of intense flux tube in the convection zone

Nonlinear interacting process of rising flux tube with convective plasma

Macroscopic observable properties of sunspots, such as latitude, tilt angle, and east-west asymmetry

(Choudhuri & Gilman 1987; D'Silva &Chouduri 1993; Fan et al. 1993; Howard 1991;Caligari et al. 1995; Fisher et al. 1995)


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Caligari, Moreno-Insertis, & Schssler (1995)simulation of rising flux tube

Ferriz-Mas (1996)

stability analysis of flux tube

Fan, Fisher, & McClymont (1994)tile angle of emerging bipolar


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There are other studies focusing on the internal structure of the buoyant flux tube

interacting with surrounding convective plasma.

Longcope, Fisher, & Arendt (1996)

Emonet & Moreno-Insertis (1998)

A rising flux tube cannot maintain its integrity unless the internal mag-

netic field is sufficiently twisted(2-dimensional MHD simulation).


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3-dimensional MHD simulation

Dorch & Nordlund (1998)

Abbett, Fisher, & Fan (2000)

... in 3-dimensional situation, the amount oftwist needed to prevent the disruption of ris-

ing flux tube is substantially reduced.Abbett, Fisher, & Fan (2000)

Dorch & Nordlund (1998)


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Photosphere:

Intermediate region between solar interior and exterior: dynamical aspect

high gas pressure region > low pressure regionthe emergence of magnetic field is a very dynamical process

thermal structure

optically thick regime > optically thin regimethe treatment of radiation is very complicated

Magara (2001)imulation of emerging flux tube

Stein & Nordlund (1998)simulation of solar granulation


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Emergence of magnetic field lines

(Color map: normal component of magnetic field on the surface)


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Physical processes of flux tube emergence (Magara 2001)

Phase I Phase II Phase III

Phase I: rising in a highly dense material (subphotpshre)

M+ mi

d2z

d t2= M mi g

vz d zdt

= M mi

M+ mig t= 4.0910 3 t

In this simulation, the flux tube almost keeps a circular cross section. > analyzing the dynamics by using the model ofa rigid cylinder rising in a gravitationally

stratified layer


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Phase II:flattening & Rayleigh-Taylor instability

convectiveunstable layer

convectivestable layer

Upper part of rising flux tube enters convective stableayer and stops rising, although lower part is still rising.

the flux tube becomes flattened!

The upper part of flux tube is subject to the Rayleigh-Tayloinstability. The dispersion relation is given by

2 = g kx

0+ 00

+ + 0 + kx

2 B0x 2

4 0+ + 0

@

i = 2

2 0

+ 1

g

4

CA2

2for> C

4 CA2

g

where p0+ = p0

+B0x

2

8 ,0

8 p0

B0x 2

, kx2

,

CA

2 p0

0 =

p0+

0+

> C


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Phase III: Parker instability (nonlinear phase)

Self-similar analysis of the nonlinear phase of Parker instability(Shibata et al. 1990)

z

s

VS

VZ

curvature radius: R

dvsdt

= d2s

dt2 g s

R

s exp t

vs = s exp t

= g/R

matter drains along the tubeaccording to the gravita-tional force.

v

z z

z 4 Bx z 1

x


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In some cases, emerging flux tube cannot expand into the atmosphere...

= C

Flattening proceeds and the

R-T unstable condition is sat-isfied, however the flux tubedoes not enter the Parker in-stability phase (phase III).

is increasing

with time

> C


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Recent development of the study of flux emergence

3-dimensional MHD simulation is now available

The plasma contained inside the flux tubedoes not drain.

The axis of flux tubehardly emerges into the atmosphere.

The plasma contained inside the flux tubedoes drain.

The flux tube becomes light,which enables the tube axis to emerge into

the atmosphere.

2-dimensional case

2-dimensional case3-dimensional case


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Matsumoto et al. (1998)...

well-developed kink state of twisted fluxtube might produce a series of sigmoidalcoronal structures.


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Magara & Longcope (2001)...

the emergence of twisted magnetic flux tube naturally forms a sigmoidal struc-ture in the atmosphere


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Distribution of vertical forces ( Pg, Pm, Tm, g )

along the outer and inner field linestime = 26

Velocity field on the outerand inner field lines


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Field-aligned velocity field (simple analytical model)

vsvss = g

Y s,

vs s = 2 g 2 b Y s sgn s

ddt

ln = vss

Strong density reduction occursin the middle of highly convex field line

at both sides of weakly convex field line

X

Y

X= a + sin

Y= b 1 + cos for

a a

2 b

g

Basic equation:

: strong densityreduction area

field line

vs


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Expanding field lines and undulating field lines

d

h

aspect ratio

h > d

aspect ratio

h < d

expanding field line

undulating field line

neutral line

neutral line


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z

force-free

gas dominant

How are emerging magnetic fields vertically stratified?

intermediate(gas dominant> magnetic dominant)

photosphere


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Injection of magnetic energy and helicity into the atmosphere

@Magnetic energy:EM t =B t

2

8 d Vz 0

@Magnetic helicity: HM t = A t B t d Vz 0

We use the concept of the relative helicity (Berger & Field 1984; Finn & Antonsen 1985;DeVore 2000)

A x,y,z, t =AC x,y, 0, t z B x,y,z

, t d z 0

z

... vector potential forB

AC x,y,z, t = z C x,y,z , t d z

z

... vector potential for the potential field

C x,y,z, t =12

Bz x ,y , 0, t

x x 2+ y y

2+z2

1 / 2d x d y

z = 0

... scalar potential for the potential field

@Magnetic energy flux in the photosphere:

FM z = 14

Bx vx +By vy Bz d x d yz = 0

+ 14

Bx2 +By

2 vz d x d yz = 0

@Magnetic helicity flux in the photosphere:

FH z = 2 Ax vx + Ay vy Bz d x d yz = 0

+ 2 Ax Bx + Ay By vz d x d yz = 0

shear term emergence term

shear term emergence term


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Time variation of magnetic energy, helicity, energy flux, and helicity flux

At the early phase, the emergence plays a dominant role in injecting energy and helicity.

At the late phase, both terms become small, however the shear term is still significant.

51027 erg

21026 erg/s

21035

erg cm

51033 erg cm /s

( 35 s) ( 35 s)

( 35 s) ( 35 s)


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Velocity field around magnetic polarity region

A rotational flow appears around peak flux area at the late phase (t = 28, 40).

This flow twists vertical magnetic field to inject energy and helicity

Rotational flow

time = 40

time = 28


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So far, the flux emergence simulation only covers the initial phase of emergence.

Simulated magnetic region: time scale... 20 min. length scale... 3000 km (footpoint

polarity region) 10,000 km (loop length)

Active region:

time scale... days length scale... 50,000 km (sunspot)

100,000 km (loop length)

A lot of important physical processes remain unclear, such as long-term evolution toward the formation of active regions interaction between emerging flux tubes and preexisting coronal fields heating, radiative cooling, thermal conduction (non-adiabatic evolution

of thermal structure)


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Interaction between emerging magnetic field and preexisting field

Simple dipole structure

Emerging field

magnetic reconnection(3-dimensional)

topological change of field mapping

rapid energy release, eruptions

Antiochos (1998) Longcope & Kankelborg (2001)

A model of energy relaxation

magnetic fluxes are transferred

from one flux domain to an-other through reconnectionprocess, which enables the re-lease of magnetic energy.


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Progress expected in the future study

To recognize when and where current sheets are formed(in 3-dimensional situation),

> use adaptive mesh technique

To follow the thermodynamical evolution,

> use dissipative energy equation (including thermal conduction, radia-tive cooling, heating process)

Yokoyama & Shibata (1997)

Courtesy of W. Abbett

t. magara- 3-dimensional evolution of a magnetic flux tube emerging into the solar atmosphere

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