THEORY OF SOLAR MAGNETIC FLUX ROPES: CMEs DYNAMICS

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THEORY OF SOLAR MAGNETIC FLUX ROPESCMEs DYNAMICS

• James Chen
• Plasma Physics Division, Naval Research Laboratory

George Mason University 21 Feb 2008
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SOLAR ERUPTIONS
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SCIENTIFIC CHALLENGES

• Observational challenges
• All remote sensing
• Different techniques observe different
  aspects/parts of an erupting structure
• (Twenty blind men )
• 3-D geometry not directly observed
• Theoretical challenges
• A major unsolved question of theoretical physics
• Energy source poorly understood
• Underlying B structure not established
• Driving force (magnetic forces) uncertain

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A FLUX-ROPE CME
LASCO C2 data 12 Sept 2000
CME leading edge (ZLE)
x
EP (Zp)
Chen (JGR, 1996)
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OBSERVATIONAL EVIDENCE
(No prominence included)

• Good quantitative agreement with a flux rope
  viewed end-on (Chen et al. 1997)
• No evidence of reconnection
• Other examples of flux-rope CMEs (Wood et al.
  1999 Dere et al., 1999 Wu et al. 1999 Plunkett
  et al. 2000 Yurchyshyn 2000 Chen et al. 2000
  Krall et al. 2001)

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OBSERVATIONAL EVIDENCE (contd)
(No prominence included)

• A flux-rope viewed from the side
• Halo CMEs are flux ropes viewed head on Krall et
  al. 2005

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3-D GEOMETRY OF CMEs

• Coronal transients (1970s OSO-7, Skylab)
• Thin flux tubes
• (Mouschovias and Poland 1978 Anzer 1978)
• Halo CMEs (Solwind) (Howard et al. 1982)
• Fully 3-D in extent
• CME morphology (SMM)
• (Illing and Hundhausen 1986)
• A CME consists of 3-parts a bright frontal rim,
  cavity, and a core
• Conceptual structure rotational symmetry (e.g.,
  ice cream cone, light bulb) (Hundhausen 1999)
• SOHO data 3-D flux ropes (Chen et al. 1997)
• 3-part morphology is only part of a CME

FOV 1.7 6 Rs
(Illing and Hundhausen (1986)
SMM (1980-1981 1984-1989)
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CME GEOMETRY WITH ROTATIONAL SYMMETRY

• Rotationally symmetric model (e.g.,
  Hundhausen1999) and some recent models
• 
  
  (SMM)
• 
  
• (1970s vintage)
  No consistent magnetic
  field

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THEORETICAL CONCEPTS TWO MODEL GEOMETRIES

• Magnetic Arcades (Traditional flare model)

Magnetic Flux Ropes

• Magnetic arcade-to-flux rope
• Energy release and formation of flux rope during
  eruption
• (e.g., Antiochos et al. 1999 Chen and Shibata
  2000 Linker et al. 2001)
• Poynting flux S 0 through the surface
• Not yet quantitative agreement with CMEs

Pre-eruption structure flux rope with fixed
footpoints (Sf) (Chen 1989 Wu et al. 1997
Gibson and Low 1998 Roussev et al. 2003)
through the surface (Chen 1989)
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GENERAL 1D FLUX ROPE

• Consider a 1-D straight flux rope
• Embedded in a background plasma of pressure pa
• MHD equilibrium
• r a minor radius
• Momentum equation
• Requirements for B
• Toroidal (locally axial) field Bt(r)
• Poloidal (locally azimuthal) field Bp(r)

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1D FLUX ROPE EQUILIBRIA

• Problem to solve specify the system environment
  pa and find a solution
• General solution
• For any given pa, there is an inifinity of
  solutions
• Must satisfy
• Problem What is the general form of B(x) in 1D?

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1D FLUX ROPE EQUILIBRIA

• Two-parameter family of solutions
• where
• Equilibrium limit p(0) gt 0

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A SPECIFIC EQUILIBRIUM SOLUTION

• Constraints Seek simplest solutions with one
  scale length (a) use the smallest number of
  terms in r/a satisfying div B 0. Demand that
  there be no singularities and that current
  densities vanish continuously at r a.
• Problem (1) Show that the following field
  satisfies div B 0.
• (2) Then find p(r), Jp(r), and Jt(r) for
  equilibrium. Note that Jp(r) must
• vanish at r 0 (to avoid a singularity)
  and r a. Therefore, Jp(r)
• must have its maximum near r/a ½.

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1D FLUX ROPE EQUILIBRIUM SOLUTIONS
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EQUILIBRIUM BOUNDARY

• Problem (1) Derive the equilibrium condition for
  given pa. Show that the equilibrium
  boundary asymptotes to the straight line
• (2) What is the physical meaning of the
  asymptote?

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A SPECIFIC EQUILIBRIUM SOLUTION

• Problem (1) Find representative solutions to
  verify the general characteristics.
• Plot the the results for different
  regimes.
• (2) Calculate the local Alfven speed inside and
  outside the current
• channel based on the total magnetic field.
  Express the results in
• terms of the ion thermal speed, assuming
  constant temperature.
• (3) Show boundary curve on the
  plot

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Example
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GOLD-HOYLE FLUX ROPE

• Repeat the same exercises using the Gold-Hoyle
  flux rope. That is, find p(r), Jp(r), and Jt(r).
• Describe the physical meaning of the parameter q?
  Define . Derive a
  constraint on in terms of q.

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TOROIDICITY

• Symmetric straight cylinder in a uniform
  background pressure
• All forces are in the minor radial direction
• Toroidicity in and of itself introduces major
  radial forces

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TOROIDICITY

• Consider a toroidal current channel embedded in a
  background plasma of uniform pressure pa
• Average internal pressure
• Toroidicity in and of itself introduces major
  radial forces
• Problem
• Show

pa
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LORENTZ (HOOP) FORCE

• Self-force in the presence of major radial
  curvature Biot-Savart law
• Consider an axisymmetric toroidal
  current-carrying plasma
• Shafranov (1966, p. 117) Landau and Lifshitz
  (1984, p. 124) Garren and Chen (Phys. Plasmas 1,
  3425, 1994) Miyamoto (Plasma Physics for Nuclear
  Fusion, 1989)

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LORENTZ (HOOP) FORCE

• Perturb UT about equilibrium (F 0)
• Assume idel MHD and adiabatic for the
  perturbations
• Total force acting on the torus
• Major radial force per unit length
• Minor radial force per unit length

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APPLICATION TO SOLAR FLUX ROPES

• Solar flux ropes
• Non-axisymmetric
• R/a is not uniform
• Eruptive phenomena
• Shafranovs derivation is for equilibrium (FR 0
  and Fa 0)
• CMEs are highly dynamic
• Drag and gravity
• Application to CMEs Assumptions

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SOLAR FLUX ROPES NON-AXISYMMETRIC GEOMETRY

• Non-axisymmetric
• Assume one average major radius of curvature
  during expansion, R
• Stationary footpoints with separation Sf
• Height of apex, Z
• However, the minor raidus cannot be
• assumed to be uniform

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SOLAR FLUX ROPES INDUCTANCE

• Inducatance L Definition
• Problem Show specific forms for the following
  choice of
• Minor radius exponentially increases from
  footpoints to apex
• Linearly increases from footpoints

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SOLAR FLUX ROPES INDUCTANCE

• Generally, as a flux rope expands,
• Problem Assume that
  Show that as a flux rope expands, its magnetic
  energy decreases as

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SOLAR FLUX ROPES INTERACTION WITH CORONA

• Dynamically, the most important interaction is
  momentum coupling (i.e., forces)
• Drag Flow around flux rope is not laminar (high
  magnetic Reynolds number)
• Gravity

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EQUATIONS OF MOTION

• Major Radial Equation of Motion
• (For simplicity, set Bct 0)
• Original derivation Shafranov (1966)
• for axisymmetric equilibrium
• Adapted for dynamics of non-axisymmetric
• solar flux ropes (Chen 1989)

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EQUATIONS OF MOTION

• Major Radial Equation of Motion
• (For simplicity, set Bct 0)
• Original derivation Shafranov (1966)
• for axisymmetric equilibrium
• Adapted for dynamics of non-axisymmetric
• solar flux ropes (Chen 1989)
• Minor Radial Equation of Motion

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COMPARISON WITH LASCO DATA

• Fits morphology and dynamics
• Fits non-trivial speed / acceleration profiles
• 11 events published Krall et al., 2001 (ApJ)

Chen et al. (2000, ApJ)
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CHARACTERISTICS OF FLUX-ROPE DYNAMICS

• Major Radial Equation of Motion

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CHARACTERISTICS OF FLUX-ROPE DYNAMICS

• Major Radial Equation of Motion

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MAIN ACCELERATION PHASE OF CMEs

• CME acceleration profiles
• Two phases of acceleration the main and residual
  acceleration phases
• The main phase Lorentz force (J x B) dominates
• The residual phase All forces are comparable,
  all decreasing with height
• A general property verified in 30 CME and EP
  events (also Zhang et al. 2001)

Calculated forces
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PHYSICS OF THE MAIN ACCELERATION PHASE

• Basic length scale of the equation of motion

• Two critical heights Z and Zm
• Z Curvature is maximum (i.e., R is minimum)
  at apex height Z (analytic result from
  geometry)
• kR(R) is maximum at
• Depends only on the 3-D toroidal geometry with
  fixed

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CHARACTERISTIC HEIGHTS

• Zm For Z gt Z, R monotonically increases
• The actual height Zmax of maximum acceleration

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EXAMPLES OF OBSERVED CME EVENTS

• Example

• CME acceleration is almost instantaneous.
• Bulk of CME acceleration occurs below 2 3 Rs
  (MacQueen and Fisher 1983 St. Cyr et al. 1999
  Vrsnak 2001)
• One mechanism is sufficient to account for the
  two-classes of CMEs Chen and Krall 2003
• Small Sf Impulsive
• Large Sf Gradual (residual
  phase)

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HEIGHT SCALES OF MAIN PHASE Observation

• Test the theory against 1998 June 2 CME
  is required

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MORE SYSTEMATIC THEORY-DATA COMPARISONS

• The Sf -scaling is in good agreement. However,
  the footpoint separation Sf is not directly
  measured Use eruptive prominences (EPs)
  for better Sf estimation

Sf 1.16 RS Zmax 0.6 RS
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DATA ANALYSIS DETERMINATION OF LENGTHS

Reverse video
Nobeyama Radioheliograph data
Sf 0.84 RS Zmax 0.55 RS
C2 data
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GEOMETRICAL ASSUMPTIONS

• The scaling law most directly given in
  quantities of flux rope (Z, Sf, Zmax, Z)
• Relationship between CME, prominence, and flux
  rope quantities
• CME leading edge (LE) ZLE Z 2aa
• Prominence LE Z Zp a
• Prominence footpoints Sp Sf 2af
• Observed quantities
• Sp, Zp, Zpmax
• Calculate
• Sf (Sp, af )
• Zmax (Zpmax, aa )

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COMPARISON WITH DATA

• CMEs Use magnetic neutral line or H filament
  to estimate Sf ZLE Z 2a
• EPs Sf Sp 2af
• Zp Z a
• 4 CMEs (LASCO)
• 13 EPs (radio, H )

• A quantitative test of the
• accelerating force and
• the geometrical assumptions
• A quantitative challenge to
• all CME models

Chen et al., ApJ (2006)
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MORE PROPERTIES OF EQUATIONS OF MOTION

• Show that during expansion the poloidal field
  (JtBp) loses energy to the kinetic energy of the
  flux rope and toroidal field.
• Some CME models invoke buoyancy as the driving
  force. Assuming Fg is the driving force,
  calculate the terminal speed of the expanding
  flux rope if the drag is force is taken into
  account. How fast can buoyancy alone drive a
  flux rope?
• Consider the momentum equation. Assuming that
  only JxB and pressure gradient force are present,
  what is the characteristic speed of a magnetized
  plasma structure of relevant dimension D? What
  is the characteristic time?
• Normalize the MHD momentum equation to the
  characteristic speed and time for plasma motions
  in the photosphere and the corona. Can the
  equations of motion distinguish between the two
  disparate mediums?