Catastrophic flux rope model for CMEs: force balance analysis and preliminary calculations of the impact of magnetic reco

nnection on the rope dynamics

Yao Chen

University of Science and Technology of China

yaochen@ustc.edu.cn

In collaboration with Prof. You-Qiu HU, Dr. Guo-qiang Li, Mr. Shu-ji Sun

Outline:(1) force balance analysis of a coronal magnetic flux rope in equilibrium or eruption

(2) preliminary calculations of the impact of magnetic reconnection on the rope dynamics

Solar Eruptive phenomena: CMEs, Flares, Prominence Eruptionsthe most violent energy release process(es) in the solar system,the dominant factor(s) disturbing the Solar-Terrestrial Space Weather

Downloaded from the SOHO website

Flux-rope catastrophe model (recent reviews: Lin et al., 2003; Hu, 2005)Flux rope: a twisted magnetic loop anchored in the photosphere

Yan et al., 2001Chen, J. 1989

Inside the rope: poloidal field (current) andaxial field (current)poloidal & axial magneticfluxes (basic rope parameters)

Observational indication

Flux rope diagram

Flux rope catastrophe:onset of a kind of global instability of the magnetic configurationa net outward resultant of magnetic forces acts on the flux rope and causes the eruption eruptive speed of the magnetic flux rope be comparable to the Alfven speed(~ 1000 km/s in the corona)

In the axisymmetric corona-flux rope system:Current distributionoutside of the rope(only azimuthal component exists)1. Bipolar background:Upper current sheetLower current sheet2. Quadrupolar bg:Upper transverse c.s.Lower vertical c.s.Inside: azimuthal/poloidal curr.

0

BJ

0

2

0 2

BBBBJ

magnetic tension

magnetic pressure

Unable to determine the contribution by a specific current to the total magnetic force acting on the flux rope

The usual way to examine the magnetic forces is todisassemble the total magnetic force into two components: the magnetic pressure and tension:

, ,r

(1) Forces by an azimuthal current

In spherical coordinates ( ), when the current density has only an azimuthal component , the magnetic vector satisfies the following Poisson equation (Lin et al., 1998)

which can be solved with the Green’s function method with given magnetic flux distribution at the lower boundary.

A 2d problem of the mag. field in spherical geometry

The solution has three parts:

(1)The corresponding potential field is determined by the given magnetic flux distribution at the lower boundary

(3)The magnetic field produced bythe image of the source current

(2)The magnetic field produced by the source current

the source current:

the image current:location: r’’=1/r’current density: - J/r’

or Ampere’s law

J : rope currentJ’: acting current (could be the source current or the image cur.)

Two equivalent methods to evaluate the forces

(1) Forces attributed to an azimuthal current

I. J azimuthal (axial) J azimuthal (axial) : Self-interaction of the rope azimuthal current(called the toroidal or hoop force by Chen, 89; curvature force by Lin et al., 98). This force is trivially zero in 2d Cartesian models by the symmetry of an infinitely long straight current)

the repulsive force by the image current

(1.1) Forces attributed to the azimuthal current inside the rope: fRa

An azimuthal current creates only poloidal fields. Their force on a poloidal current is in the azimuthal direction, and should be zero because of axisymmetry. no force on the rope poloidal current

II. J’ azimuthal (axial) J azimuthal (axial)

(2) Forces attributed to a poloidal current For the axisymmetric system the poloidal current and the consequent azimuthal magnetic field exist only inside the rope (can be proven). has no corresponding image current exerts no forces on any azimuthal current

(on the rope per radian in azimuthal width)

The self-interaction of the rope poloidal current:

(3) Forces attributed to the corresponding bk potential field Dipole potential field

( in units of B0)Background: partially open bipolar field

quadrupolar bg potential field(Antiochos et al., 1998)

(III) Forces attributed to the rope currentsJ (J’)azimuthal (axial) J azimuthal (axial) : fRa

J poloidal J poloidal : fRp (self-interaction)

Classifications of magnetic forces acting on the rope currents: (I) Force produced by the bk potential field : fp

(II) Forces associated with the current in the current sheets: upper, lower, transverse: fc1,c2,c3

Direction of the forces on the rope unit: radial• Upward lifting force • Downward pulling force (confining force or restoring force)

X

photosphere Flux rope

Aim of the force balance analysis:to analyze the interplay among the different pieces of magnetic forces which play dominant roles in the equilibrium and eruptive process of the flux rope in a variety of field topologies.

fRa :

fRp :

fp :

fc1 :

∑f :

Force-free field in equilibrium: partially-open bipolar bg field

Poloidal fluxPoloidal flux

fRa

the rope azimuthal current: self-force + the repulsive force by the image the rope poloidal current: self-force the background potential fieldthe current in the upper current sheetThe resultant of forcesForce unit:

Main points: (1)The plots are obtained for a series of equilibrium force-free solutions with different rope fluxes(2) ∑f < 0.003 : The force-free condition is well satisfied(3) Dominant forces maintaining the rope in equilibrium: fRa .vs. fp

(4) Other forces with much smaller amplitude do not vary significantly with the rope flux

(5) Catastrophe sets in with a slightly larger poloidal flux

Poloidal fluxPoloidal flux

fRa

Meta-stable equilibrium state

the rope azimuthal currentthe current in the upper current sheet (dashed)the bg potential field (solid)the rope poloidal current: self-force (NEW)the current in the newly-formed lower c.s.The resultant of forces

fRa :fc1 :-fp :fRp :fc2 :∑f :

Rope eruption (MHD solution ): partially-open bipolar bg fieldfRa

1.Variation of the main forces, 2. newly-formed c.s. 3. the resultant of forces

Main points:(1)Dominant lifting or

driving force fRa main pulling forces: fp + fc2

(2) fc2 becomes the dominant pulling force after t=60 minutes

(3)Resultant of forces: upwarderuption of the rope

Rope topRope axisRope bottom

fRa

fRa :

fRp

:- fp :

fc3 :

∑f :

the rope azimuthal current: self-force + the repulsive force by the image the rope poloidal current: self-force the bg potential field the current in the upper c.s.The resultant of forces

Force-free field in equilibrium: quadrupolar bg field

Poloidal fluxPoloidal flux

Poloidal fluxPoloidal flux

Main points: (1)The solutions are obtained for a series of equilibrium solutions with different rope fluxes(2) ∑f < 0.01 : The force-free condition is well satisfied(3) Dominant forces maintaining the rope in equilibrium: fRa .vs. fp

(4) Other forces do not vary significantly with the rope flux(5) Catastrophe takes place with a slightly larger poloidal flux

Meta-stable equilibrium state

the rope azimuthal current:self-force + the repulsive force by the imagethe rope poloidal current: self-force the bg potential fieldthe current in the transverse c.s.(NEW)the current in the newly-formed lower vertical c.s.The resultant of forces

fRa :

fRp : fp :fc3 :fc2 :

∑f :

Rope eruption (MHD solution ): quadrupolar bg field

fRa

1.Variation of the main forces, 2. newly-formed c.s. 3. the resultant

Main points:(1)Dominant lifting or

driving force fRa main pulling or resto

ring forces: fc2 + fc3

(2) fc2 fc3 become the dominant pulling forces after t=55 minutes

(3) fp changes direction (4) Resultant of forces:

upwarderuption of the rope

Summary:(1) In the coronal-flux rope system in equilibrium:The dominant lifting force is attributed to the azimuthal current inside the rope.The dominant pulling force is attributed to the background potential field.(2) for the eruptive flux rope after catastrophe: Driving/lifting force the azimuthal current inside the ropeMain restoring/pulling force the current in newly-formed c.s.

About the effect of magnetic reconnection on CME dynamics(1) the Ohm’s dissipation in the reconnection site (heats & accelerates the plasmas, thermal pressure)(2) An enhanced outward lifting Lorentz force resulted by the erosion of the current in the current sheet (Maybe more important than the first aspect) (see also Low & Zhang, 2002 … ).

• polytropic solar wind (γ=1.05)

• ideal MHD V.S. resistive MHD

Effect of magnetic reconnection on the rope dynamics: preliminary calculations

Ideal MHD:Special measure to avoid/prohibit numerical reconnections(trick: usage of the magnetic flux function as a basic variant to prohibit reconnectionslong curr. sheet)

Resistive MHD:Homogeneous anomalous resistivity

Results of ideal & resistive MHD calculations:

Color countors: velocity

0000 sv01.00

Comparison between the idealand resistive MHD calculations.(t=200 minutes)

Resistive

Ideal Rapid expansion with fast eruption!

Cusp pointRope topRope axisRope bottom

B0=4G

Resistive

Ideal

B0=8GCusp pointRope topRope axisRope bottom

B0: field atrength at the base of the equator

Resistive

Ideal

B0=4G

B0=8G

Comparison with observations:Zhang et al., 2004, ApJ

Cusp pointRope topRope axisRope bottom

1. Velocity profile and value2. Time taken for the main acceleration: 2 hours

Preliminary conclusions:

I: Varying B0 (magnetic field strength at the base along the equator)a smooth transition from fast to slow CMEs (Are there really two types of CMEs?)II: Magnetic reconnections have significant impacts on CME speed by reducing/eliminating the pulling force of the current sheets.

Future study:• investigate the initiation and propagation of the rope in a background with more realistic solar wind conditions• quantitative analysis on the magnetic free-energy released by magnetic reconnection and MHD catastrophe, the distribution of the released energy among the thermal and kinetic energies, the roles played by Ohm's dissipation and the work done by the Lorentz force• CME-driven shock properties, relevant particle acceleration with a combination of the MHD calculation and a kind of kinetic model (?)

Thanks!

In the axisymmetric (2.5d) flux rope system:outside of the rope:Currents concentrate in the current sheets & flow in the azimuthal direction; inside the rope: both azimuthal (or axial) and poloidal components are present.

Numerical example of the flux rope eruption after catastrophe that is triggered by a very small adjustment of the rope flux(background field:a partially-open bipolar field with an c.s.)