3d simulations of solar emerging flux
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3D simulations of solar emerging flux. ISOBE Hiroaki Plasma seminar 2004/04/28. What is emerging flux?. Magnetic field: origin of solar activities such as sunspots, active regions, coronal heating, flares, coronal mass ejections, jets etc..

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3D simulations of solar emerging flux

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3D simulations of solar emerging flux

ISOBE Hiroaki

Plasma seminar 2004/04/28

What is emerging flux?

  • Magnetic field: origin of solar activities such as sunspots, active regions, coronal heating, flares, coronal mass ejections, jets etc..

  • Magnetic field is generated by the dynamo action in the solar interior and rise to the surface by magnetic buoyancy.

  • Dynamics of the emergence of the magnetic flux into the atmosphere is impotant to understand: formation of sunspots of active region, energy accumlation and triggering of flares, jets and other ecplosive phenomena.

Shibata et al. 1989



2D simulation

Parker instability =>Expansion due to magnetic pressure => formation of Ω-shaped loop in the corona.

3 dimensionality

  • Evidence for emergence of twisted flux tube such as Sigmoid.

Matsumoto et al. 1993

  • 3D, fully compressible, ideal MHD

  • Noliner development of interchange and undular (Parker) mode

  • Extraction of the gas from the loop top by the undular mode is necessary for the emergence into the corona.

Matsumoto et al. 1998

Fan 2001 (ApJL)

Magara & Longcope 2001

  • Fully compressible, ideal MHD

  • Emergence of twisted flux tube

  • Outer loop is close to potential, while inner loop exhibits sigmiod-like structure.

Magara & Longcope 2003

Measurement of MHD helicity injection by the emerging flux using the method presented by Kusano et al. (2002)

A Model for Dynamic Evolution of Emerging Magnetic Field in the Sun Magara, T 2004, ApJ, 605, 480

  • 3D simulation of twisted flux tube

  • Modeling of the height evolution of field line with various shape.

Model setup

*unit of legnth = 2×scale height at z=0

  • Ideal MHD. Hydrostatic initial atmoshere, twisted flux in the convection zone.

  • Grid: 100x100x100 (only 1/4 of the domain is solved)

Snapshot of outer/inner field lines

  • Outer filed line expand like a wide fan.

  • Inner field line extends mainly in the vertical direction and keeps parallel to the original axis.

Mechanism of different behavior of outer/inner filed lines

The inner field line cannot expand laterally because it is surrounded tightly by adjacent twisted field lines, while the outer field line can expand freely.

Modeling of emergence of field line

To model the height evolution of the apex of the filed line, the momentum equation for the plasma element at the middle of the emerging filed line is solved.





Derivation of equation 1. density profile

Derivation of equation 2. JxB force

Magnetic field strehgth B is approximated to have z dependance like eq.(19)

Derivation of equation 3.

↑equation to be solved, where vAi=Bi/(4πρi), κ:curvature of the fied line, H: scale height of B.

H, zi, and Bi are obtaine by fitting of the simulation reslut:

H=21, zi=10, Bi=0.13

Result for outer field line

Solution of the

model equation.

Result of simulation.

Result for different types of field line

typeA: constant κ

typeB: κincrease with time

typeC: κdecrease with time



Type A continues to expand. Type B approaches to quasi-static state. Density is larger and tension is less important in type C: good model for prominence?

Chirality of field lines

If initial twist is left-handed, inner field line (dark gray) exhibits backward-S shape, while outer fied line (light gray) exhibits S shape.

Whichfield lines are brighten?

Field lines with stronger current at their footpoints are drawn in lighter color.

Inner field lines are bright and backward-S is seen.

The Emergence of a Magnetic Flux Tube into a Preexisting Coronal ArcadeFan, Y. & Gibson, S.E., 2003, ApJ, 589, L105

  • 3D isothermal (lowβ) MHD simulation of emerging flux tube into the corona with pre-existing magnetic arcade.

  • Emerging flux strongly twisted and kink-unstable.

  • The emergence is driven by the electric field at the lower boundary.

Model setup

  • Potential arcade in the initial corona.

  • Resistivity is numerical.

  • Emergence of twisted flux is driven by the electric fiedl at the lower boundary.

  • grid: 240x160x200


Rapid rise starts after kink instability

sets in.

Result (2)

  • Strong current sheet is formed and exhibits inverse-S

A Coupled Model for the Emergence of Active Region Magnetic Flux into the Solar CoronaAbbett, W.P. & Fisher G.H., 2003, ApJ, 582, 475

  • 3D MHD simulation of emerging magnetic flux tube from convection zone into the corona.

  • The convection zone and upper atmosphere is solved separately with different scheme, and the result of the convection zone calculation is used as the lower boundary of the upper calculation domain.

  • (The numerical technique is interesting, the presented results are not so much.)

Simulation domains

Photosphere-corona: Fully compressive MHD (ZEUS 3D).

64x64x64 grids (1grid=1Mm).


Convection zone: Anelastic MHD (ANMHD; Lantz & Fan 1999).

256x128x128 grids (1grid=1Mm).

Adiabatically stratified, depth=5.1 pressure scale heights.

Result (1)

MHD Potential field

No twist

  • Potential approximation is not good when the flux tube twisted (not surprising).

  • But the field are not force-free even in the case of q=0.



Result (2)

blue: θ=0

(force free)

black: θ=π/4

magenta: θ= π/2

θ: angle from potensial field.

Outer field lines differs more from force-free (hence more dynamic).


Result (3)

After the vertical flow at the lower boundary diminished (no driving at the boudary), the field lines relax to more force-free.


Result (4)

Some field lines exhibit sigmoidal structure. But the chirality depends on field lines.

My simulations 1.With convection

  • Emergence of twisted flux tube from vigorously convecting convection zone

  • Emerging flux loses its coherence by the turbulent convective flow.

Reconnection with pre-existing field

  • 3D version of Yokoyama & Shibata 1995

  • Fast reconnection by anomalous resistivity=> jet formation.

Formation of 3D structure by magnetic Rayleigh-Taylor instability

Effect of lateral expansion

Confinement by strong coronal field (Miyagoshi-san’s calculation).

λ= 5 10 100


If λis small, convective interchange mode and lataral expansion prevent the emergence into the corona

2 dimensional

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