NATO Advanced Research Workshop 3-dimensional Evolution of a Magnetic Flux Tube Emerging into the Solar Atmosphere

T. Magara (Montana State University, USA)

September 17, 2002 (Budapest, Hungary) We focus on three solar regions.

Each region has a different type of background gas layers with which magnetic field interacts.

¥ The corona (over the photosphere) ... low-density background gas layers

¥ The photosphere ... abrupt change of background gas layers

¥ The convection zone ... high-density background gas layers 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 in such 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) 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.

Input stage: impose various type of perturbations to the system according to the primary purpose of studies

perturbation Initial stage: set a skeleton of magnetic field in the atmosphere (eg. potential, force-free, or relaxation state: any magnetohydrostatic To see the thermal and states) dynamical evolutions at the linear and nonlinear phases

magnetic reconnection Convection zone: Gas dominant region Ð> 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(Choudhuri & Gilman 1987; D'Silva & Chouduri 1993; Fan et al. 1993; Howard 1991; ¥ Stability of intense flux tube in the convection zone Caligari et al. 1995; Fisher et al. 1995) ¥ Nonlinear interacting process of rising flux tube with convective plasma ¥ Macroscopic observable properties of sunspots, such as latitude, tilt angle, and east-west asymmetry Ferriz-Mas (1996) ‘stability analysis of flux tube’ Caligari, Moreno-Insertis, & Schüssler (1995) ‘simulation of rising flux tube’

Fan, Fisher, & McClymont (1994) ‘tile angle of emerging bipolar’ 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). 3-dimensional MHD simulation

Dorch & Nordlund (1998) Abbett, Fisher, & Fan (2000)

... in 3-dimensional situation, the amount of twist needed to prevent the disruption of ris- ing flux tube is substantially reduced. Abbett, Fisher, & Fan (2000)

Dorch & Nordlund (1998) Photosphere:

Intermediate region between solar interior and exterior: ¥ dynamical aspect high gas pressure region Ð> low pressure region the emergence of magnetic field is a very dynamical process ¥ thermal structure optically thick regime Ð> optically thin regime the treatment of radiation is very complicated

Magara (2001) imulation of emerging flux tube

Stein & Nordlund (1998) simulation of solar granulation Emergence of magnetic field lines (Color map: normal component of magnetic field on the surface) Physical processes of flux tube emergence (Magara 2001)

Phase I Phase II Phase III

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

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

2 M + m d z =Ð M Ð m g i dt2 i ↓

≡ dz M Ð mi × Ð3 vz =Ð gt= 4.09 10 t dt M + mi Phase II: flattening & Rayleigh-Taylor instability

convective Upper part of rising flux tube enters ‘convective stable stable layer ayer’ and stops rising, although lower part is still rising. ↓ convective unstable layer the flux tube becomes flattened!

The upper part of flux tube is subject to the Rayleigh-Taylo instability. The dispersion relation is given by λ λ > C + Ð Ð ρ Ð ρ B 2 ω2 =Ðgk 0 0 + k2 0 x x ρ+ + ρÐ x πρ+ ρÐ 0 0 4 0 + 0 ↓

2 π 2 ω 2 π g π C A λ λ ≡ 4 C A i = βÐ λ Ð4 2 for > C g 2 0 +1 λ

Ð 2 Ð B 8 π p k where p+ = pÐ + 0 x , βÐ ≡ 0 , λ≡ x , 0 0 8 π 0 Ð 2 2 π B0 x pÐ p+ 2 ≡ 0 0 C A ρÐ = ρ+ 0 0 Phase III: Parker instability (nonlinear phase)

Self-similar analysis of the nonlinear phase of Parker instability (Shibata et al. 1990) dv 2 z s = d s ≈ g s dt dt2 R

VZ matter drains along the tube according to the gravita- tional force. VS

s s ∝ exp Ω t Ω ∝ Ω vs = s exp t x Ω = g / R curvature radius: R

∝ vz z

Ð4 ∝ Ð1 ρ∝ z Bx z In some cases, emerging flux tube cannot expand into the atmosphere...

λ λ > C λ λ = C

Flattening proceeds and the λ is increasing λ with time R-T unstable condition is sat- isfied, however the flux tube does not enter the Parker in- stability phase (phase III). The reason of no expansion is that the magnetic pressure of flux tube is weaker than surrounding gas pressure. z expansion case no expansion case

Emerging magnetic fields do not always expand to form a well-developed coronal structure. If they are magnetically strong, they can expand, other- wise the emerging fields could be easily controlled by strong photospheri motions so that they show an intermittent behavior in the photosphere. Recent development of the study of flux emergence 3-dimensional MHD simulation is now available

2-dimensional case

2-dimensional case 3-dimensional case The plasma contained inside the flux tube The plasma contained inside the flux tube does not drain. does drain. ↓ ↓ The axis of flux tube The flux tube becomes light, hardly emerges into the atmosphere. which enables the tube axis to emerge into the atmosphere. Matsumoto et al. (1998)...

well-developed kink state of twisted flux tube might produce a series of sigmoidal coronal structures. Fan (2001)

The axis of flux tube can emerge into the atmosphere.

the magnetic field and velocity field resulted from the emergence of twisted flux tube are consistent with observational results. Magara & Longcope (2001)... the emergence of twisted magnetic flux tube naturally forms a sigmoidal struc- ture in the atmosphere ∇ ∇ ρ Distribution of vertical forces (Ð Pg,Ð Pm, Tm, g) along the outer and inner field lines

time = 26

Velocity field on the outer and inner field lines Field-aligned velocity field (simple analytical model)

Y Basic equation: ∂ v ∂ ∂v v s =Ðg Y, d ln ρ =Ð s g s ∂s ∂ s dt ∂s

vs

2 b vs s =2g 2 b Ð Ys sgn s field line Ð π a π a X

X = a θ + sin θ for Ð π≤θ≤π Y = b 1 + cos θ : strong density reduction area

Strong density reduction occurs

in the middle of highly convex field line at both sides of weakly convex field line Expanding field lines and undulating field lines

aspect ratio h > d expanding field line

neutral line h d aspect ratio h < d

undulating field line

neutral line How are emerging magnetic fields vertically stratified?

force-free

z intermediate (gas dominant gas dominant Ð> magnetic dominant)

photosphere Injection of magnetic energy and helicity into the atmosphere

Bt2 Magnetic energy: E t = dV M 8 π z ≥ 0 ⋅ Magnetic helicity: HM t = A t B t dV z ≥ 0

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

z × ′ ′ A x, y, z, t = AC x, y,0,t Ð z B x, y, z , t dz... vector potential for B 0 ∞ ∇× φ ′ ′ AC x, y, z, t = z C x, y, z , t dz... vector potential for the potential field z

B x′, y′,0,t φ 1 z ′ ′ C x, y, z, t = π 1/2 dx dy... scalar potential for the potential field 2 2 2 x Ð x′ + y Ð y′ + z2 z =0

Magnetic energy flux in the photosphere: 1 1 2 2 FMz=Ðπ Bx vx + By vy Bz dxdy+ π B x + B y vz dxdy z =0 4 z =0 4 shear term emergence term

Magnetic helicity flux in the photosphere:

FHz=Ð2A x vx + A y vy Bz dxdy+2A x Bx + A y By vz dxdy z =0 z =0 shear term emergence term Time variation of magnetic energy, helicity, energy flux, and helicity flux

× 2 ⋅1026 erg / s

× 5 ⋅1027 erg

(× 35 s) (× 35 s)

× 2 ⋅1035 erg cm

× 5 ⋅1033 erg cm / s

(× 35 s) (× 35 s) 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. Velocity field around magnetic polarity region

Rotational flow

time = 40

time = 28

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

Simulated magnetic region: Active region: ¥ time scale... 20 min. ¥ time scale... days ¥ length scale... 3000 km (footpoint ¥ length scale... 50,000 km (sunspot) polarity region) 100,000 km (loop length) 10,000 km (loop length) Interaction between emerging magnetic field and preexisting field

Simple dipole structure magnetic reconnection (3-dimensional) ↓ topological change of field mapping ↓ rapid energy release, eruptions Emerging field

A model of energy relaxation

magnetic fluxes are transferred from one flux domain to an- other through reconnection process, which enables the re- lease of magnetic energy. Antiochos (1998) Longcope & Kankelborg (2001) Progress expected in the future study

¥ To recognize when and where current sheets are formed (in 3-dimensional situation), Ð> use adaptive mesh technique

Courtesy of W. Abbett

¥ To follow the thermodynamical evolution, Ð> use dissipative energy equation (including thermal conduction, radia- tive cooling, heating process)

Yokoyama & Shibata (1997)