Jan 13, 2009 ISSI 1

Modeling Coronal Flux Ropes

A. A. van BallegooijenHarvard-Smithsonian Center for Astrophysics,

Cambridge, Massachusetts, U.S.A

Collaborators: M. Bobra, S. Cranmer, Y. Su

Jan 13, 2009 ISSI 2

Flux Ropes in Active RegionsBobra et al. (2008) constructed non-linear force-free field (NLFFF) models of an active region (NOAA 9997/10000), based on an observed TRACE loop (panel b). Models are constructed by inserting a flux rope into a potential field, then applying magneto-frictional relaxation.

Model #8 (best fit to observations)

Left panels show the flux rope (overlying arcade not shown).

Model parameters:Axial flux:Φaxi = 4×1020 Mx (W section)Φaxi = 14×1020 Mx (E section)Poloidal flux:Fpol = -1010 Mx/cm.

Jan 13, 2009 ISSI 3

Flux Ropes on the Quiet Sun

Cavities observed in X-rays are thought to be due to coronal magnetic flux ropes:

Cavity

Jan 13, 2009 ISSI 4

Flux Ropes on the Quiet SunGoal is to apply flux-rope insertion method to the quiet Sun (coronal cavities, filament channels). Here I present initial results for 2007 April 12-19:

Polar CH

Low-LatitudeCoronal Hole

Jan 13, 2009 ISSI 5

Part I: Modeling Coronal Flux Ropes

Starting from full-disk magnetogram (SOLIS), a latitude-longitude map of the radial field is constructed. This provides the lower boundary condition for the 3D magnetic model.

Note: both coronal holes have positive polarity (white).

Jan 13, 2009 ISSI 6

Modeling Coronal Flux RopesThere are two polarity inversion lines. The H image (Kanzelhöhe) from Apr 19 shows only a small filament fragment:

Red=positive,Green=negative

Jan 13, 2009 ISSI 7

Modeling Coronal Flux Ropes

This filament fragment is observed as a prominence with Hinode/SOT (left) and XRT (right) on April 25, 2007 (Heinzel et al. 2008):

Jan 13, 2009 ISSI 8

Modeling Coronal Flux Ropes

Despite the absence of long H filaments, I manually select two flux rope paths.Two artificial sources (± 1020 Mx) have been added at the left ends.

Red=positive,Green=negative

Jan 13, 2009 ISSI 9

Modeling Coronal Flux Ropes

Compute potential field (partial sphere geometry, Rmax = 2.4 Rs), create “cavities” above selected paths, and insert sinistral flux ropes (axial flux 1020 Mx; poloidal flux 5x109 Mx/cm):

Selected field lines after only 1000 iterations. Overlying arcades not shown.

Then apply magneto-frictional relaxation for 20,000 iterations.

Jan 13, 2009 ISSI 10

Modeling Coronal Flux Ropes

Spatial distribution of current-helicity (x,y,z) in the NLFFF after 20,000 iterations:

Jan 13, 2009 ISSI 11

Modeling Coronal Flux Ropes

Side view of northern flux rope ( in vertical plane along PIL).There is a current layer between the flux rope and its surroundings.Note the “waviness” of this current layer.

Jan 13, 2009 ISSI 12

Modeling Coronal Flux Ropes

The same waviness is seen in a second model with reduced axial- and poloidal fluxes:

It is due to the break-upof the current layer into channels with twisted field lines:

Jan 13, 2009 ISSI 13

Modeling Coronal Flux Ropes

Field-line dips (blue) occur in disconnected patches everywhere along PIL,not just at observed filament fragment (prominence):

Jan 13, 2009 ISSI 14

Modeling Coronal Flux Ropes

Selected field lines in the flux ropes (left) and overlying arcades (right) in the first model, rotated 60:

In both models the eastern part of the northern flux rope is unstable, probably due to the lack of an overlying arcade.

Jan 13, 2009 ISSI 15

Modeling Coronal Flux Ropes

Reconnection between northern flux rope (left) and polar field (far right) in eastern part of first model:

Jan 13, 2009 ISSI 16

Part II: Coronal HeatingThere are two sources of energy for coronal heating:

• Energy may propagate into the corona from the convection zone. Parker (1972) proposed that the corona is heated by twisting/braiding of magnetic field lines due to small-scale, random footpoint motions.

• Energy may already be stored in the corona. Coronal flux ropes contain large amounts of magnetic free energy. Some of this energy may be converted into heat (van Ballegooijen & Cranmer 2008).

Jan 13, 2009 ISSI 17

Coronal HeatingThe idea is that the magnetic fields in a coronal flux rope are to some degree stochastic (Lazarian & Vishniac 1999).

Resistive MHD turbulence within the flux rope converts mean magnetic energy into heat via reconnection.

The process can be described in terms of hyperdiffusion, a type of magnetic diffusion in which the magnetic helicity is conserved (Boozer 1986; Bhattacharjee & Hameiri 1986) :

( ). 02042

0

000

0 αη ∇⋅∇+×=∂

∂B

B

BBv

t

Ar

rrr

Jan 13, 2009 ISSI 18

Coronal HeatingIn the model by van Ballegooijen & Cranmer (2008), the total heating rate ε is a sum of a direct contribution from footpoint motions and a contribution from hyperdiffusion:

where L is the loop length.

Parameters: uph = 0.35 km/s, τph = 600 s, Bph = 1500 G, λturb = 103 km.

Modeling approach:• construct NLFFF model, α0(r)• compute heating rate ε• compute temperature and density

.||)(4

105)(4

3)( 20

4turb

2ph3

02

ph2phph

πλ

πτ

ε ∇×+≈ − svL

BsB

L

uBs A

Jan 13, 2009 ISSI 19

Coronal HeatingTemperature (left) and density (right) for model with hyperdiffusion, in vertical cross-section of the flux rope (y=0):

Jan 13, 2009 ISSI 20

Coronal HeatingTemperature (left) and density (right) for model without hyperdiffusion:

Jan 13, 2009 ISSI 21

Summary

Developed preliminary models for flux ropes on polar crown and sub-polar crown channels based on observed photospheric fields (SOLIS). Remains to be seen whether surrounding arcades can hold down the flux rope(s).

Modeled the heating in coronal flux ropes, including the effect of hyperdiffusion.Depending on the spatial distribution of α, such models may explain the coronal cavities associated with flux ropes.