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

2. 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

3. Solar Eruptive phenomena: CMEs, Flares, Prominence Eruptions the 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

4. 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., 2001 Chen, J. 1989 Inside the rope: poloidal field (current) and axial field (current)  poloidal & axial magnetic fluxes (basic rope parameters) Observational indication Flux rope diagram

5. Flux rope catastrophe: onset of a kind of global instability of the magnetic configuration  a 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)

6. In the axisymmetric corona-flux rope system: Current distribution outside of the rope (only azimuthal component exists) 1. Bipolar background: Upper current sheet Lower current sheet 2. Quadrupolar bg: Upper transverse c.s. Lower vertical c.s. Inside: azimuthal/poloidal curr.

7.  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 to disassemble the total magnetic force into two components: the magnetic pressure and tension:

8. (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

9. 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 by the 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’

10. or Ampere’s law J : rope current J’: 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

11. 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: f Ra 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)

12. (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:

13. (3) Forces attributed to the corresponding bk potential field Dipole potential field ( in units of B 0 ) Background: partially open bipolar field quadrupolar bg potential field (Antiochos et al., 1998)

14. (III) Forces attributed to the rope currents J (J’)azimuthal (axial) J azimuthal (axial) : f Ra J poloidal J poloidal : f Rp (self-interaction) Classifications of magnetic forces acting on the rope currents: (I) Force produced by the bk potential field : f p (II) Forces associated with the current in the current sheets: upper, lower, transverse: f c1,c2,c3 Direction of the forces on the rope unit: radial Upward lifting force Downward pulling force (confining force or restoring force) X photosphereFlux rope

15. 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.

16. f Ra : f Rp : f p : f c1 : ∑f : Force-free field in equilibrium: partially-open bipolar bg field Poloidal flux f Ra  the rope azimuthal current: self-force + the repulsive force by the image  the rope poloidal current: self-force  the background potential field  the current in the upper current sheet  The resultant of forces Force unit:

17. 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: f Ra.vs. f p (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 flux f Ra Meta-stable equilibrium state

18.  the rope azimuthal current  the 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 f Ra : f c1 : -f p : f Rp : f c2 : ∑f : Rope eruption (MHD solution ): partially-open bipolar bg field f Ra 1.Variation of the main forces, 2. newly-formed c.s. 3. the resultant of forces

19. Main points: (1)Dominant lifting or driving force f Ra main pulling forces: f p + f c2 (2) f c2 becomes the dominant pulling force after t=60 minutes (3)Resultant of forces: upward  eruption of the rope Rope top Rope axis Rope bottom

20. f Ra f Ra : f Rp : - f p : f c3 : ∑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 flux

21. 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: f Ra.vs. f p (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

22.  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 transverse c.s.  (NEW)the current in the newly-formed lower vertical c.s. The resultant of forces f Ra : f Rp : f p : f c3 : f c2 : ∑f : Rope eruption (MHD solution ): quadrupolar bg field f Ra 1.Variation of the main forces, 2. newly-formed c.s. 3. the resultant

23. Main points: (1)Dominant lifting or driving force f Ra main pulling or restoring forces: f c2 + f c3 (2) f c2 f c3 become the dominant pulling forces after t=55 minutes (3) f p changes direction (4) Resultant of forces: upward  eruption of the rope

24. 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 rope Main 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 … ).

25. 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 reconnections  long curr. sheet) Resistive MHD: Homogeneous anomalous resistivity

26. Results of ideal & resistive MHD calculations: Color countors: velocity

27. Comparison between the ideal and resistive MHD calculations. (t=200 minutes) Resistive Ideal Rapid expansion with fast eruption! Cusp point Rope top Rope axis Rope bottom

28. B 0 =4G Resistive Ideal B 0 =8G Cusp point Rope top Rope axis Rope bottom B 0 : field atrength at the base of the equator

29. Resistive Ideal B 0 =4G B 0 =8G Comparison with observations: Zhang et al., 2004, ApJ Cusp point Rope top Rope axis Rope bottom 1.Velocity profile and value 2.Time taken for the main acceleration: 2 hours

30. Preliminary conclusions: I: Varying B 0 (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.

31. 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 (?)

32. Thanks!

33. 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.)