theory of solar magnetic flux ropes: cmes dynamics james chen plasma physics division, naval...
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THEORY OF SOLAR MAGNETIC FLUX ROPES:
CMEs DYNAMICS
James ChenJames ChenPlasma Physics Division, Naval Research LaboratoryPlasma Physics Division, Naval Research Laboratory
George Mason University 21 Feb 2008
SOLAR ERUPTIONS
• Canonical parameters of solar eruptions:
– KE, photons, particles erg– Mass g– Speed ~100 - 2000 km/s
• Space Weather: CMEs are the solar
drivers of large geomagnetic storms
– Physical mechanism: CMEs lead to
strong-B solar wind (SW) structures
(i.e., strong southward IMF) impinging
on the magnetosphere
– CMEs are magnetically dominated
structures
32 33~ 10
14 16~ 10
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
SCIENTIFIC CHALLENGES
• SOHO new capabilities and insight
90
OBSERVATIONAL EVIDENCE
• 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)
(No prominence included)
OBSERVATIONAL EVIDENCE (cont’d)
• A flux-rope viewed from the side
• Halo CMEs are flux ropes viewed head on [Krall et al. 2005]
(No prominence included)
3-D GEOMETRY OF CMEs
• “Coronal transients” (1970’s: 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
SMM (1980-1981; 1984-1989)
(Illing and Hundhausen (1986)
• Rotationally symmetric model (e.g., Hundhausen1999) and some recent models
(SMM)
(1970s vintage) No consistent magnetic field
CME GEOMETRY WITH ROTATIONAL SYMMETRY
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)0S
0S
0S
• 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)
GENERAL 1D FLUX ROPE
J B1 x 0c p
0 B
221 1
8 4pBp B
r r r
pa
1/22 2( , ) , ,t p t pr B B B B B B
• 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?
1D FLUX ROPE EQUILIBRIA
22 2
0
1 1( ) (0) ( ) (0) , ( )
8 4
rp
tB
p r p B r B dr r ar
22 2
0
1 1( ) (0) ( ) (0) , ( )
8 4
ap
p t aB
p r p B a B dr p r ar
B J B0 and
4
c
• Two-parameter family of solutions;
where
• Equilibrium limit: p(0) > 0
1D FLUX ROPE EQUILIBRIA
2
2* *
0
( )( )1 11 2 (0) 1, ( ) , ( )
ap pt
t t pt pap t
b B rB rdr b b r b r
r B B
* *2 *2 2 *2
8 81 1,a a
t pt t pa p
p p
B B B B
2
0
( ), 2 ( ) /a
pa p t tB B a B B r rdr a
• 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.
A SPECIFIC EQUILIBRIUM SOLUTION
( )tB r 2 4ˆ ˆ ˆ3 1 2 , / 1tB r r r r a
ˆ0 1r
( )pB r 2 4ˆ ˆ ˆ ˆ3 1 / 3 , 1paB r r r r
1ˆ ˆ, 1paB r r
1D FLUX ROPE EQUILIBRIUM SOLUTIONS
(0) ap p
0 (0) ap p
* *
1/2 * 1/2 *
1 1,
8 8
patt p
t pa a
BBB B
p p
• 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?
EQUILIBRIUM BOUNDARY
1/2* *90
47p tB B
• 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
A SPECIFIC EQUILIBRIUM SOLUTION
1/22 2( ) ( )( )
4t p
Ai
B r B rV r
nm
( ) 0 1 * *p tB B
• 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.
GOLD-HOYLE FLUX ROPE
02 2
( ) ,ˆ1
tB
B rq r
02 2
ˆ( )
ˆ1p
B qrB r
q r
1/2* 20 1B q
*0B
• 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
TOROIDICITY
pa
• 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
TOROIDICITY
pa
p
p
2
0 0 0cos( ) cos
ap
dpF dr d d r R r
dr
2 / 8a
p ppa
p pF
B
• 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)
LORENTZ (HOOP) FORCE
84 ln 2
2iR
L Ra
22 2 21
2 , ,8 2
t ctt p t T t p
B BU Ra U LI U U U
F J B J B
1,
4
cp
c
F ξ x 3TU d
x
23
8 1TB p
U dR
ap,c ctB B
• 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:
LORENTZ (HOOP) FORCE
R
ap
,c ctB B
F R̂2T RF R
2 22
2 2
8 1 1ln 2 1
2 2 2t ct c it
R ppp
B BI R R BF
a a Bc R B
2 / 8a
ppa
p p
B
2 2
2 21t t
a ppa
I BF
c a B
• Solar flux ropes:
– Non-axisymmetric
– R/a is not uniform
• Eruptive phenomena:
– Shafranov’s derivation is for equilibrium (FR = 0 and Fa = 0)
– CMEs are highly dynamic
– Drag and gravity
• Application to CMEs: Assumptions
APPLICATION TO SOLAR FLUX ROPES
• 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
SOLAR FLUX ROPES: NON-AXISYMMETRIC GEOMETRY
2 2 / 4
2fZ S
RZ
( )
R R
a a
• Inducatance L: Definition
• Problem:: Show specific forms for the following choice of
– Minor radius exponentially increases from footpoints to apex
– Linearly increases from footpoints
SOLAR FLUX ROPES: INDUCTANCE
x2 3 21 1
8 2p p tU B d LI
( ) exp( )fa
1 8 8
ln ln 22 2
i
f a
R R
a aL
2
4,
RL
cL
( ) fa
2
4 1, 1 ln , /
2 1 a fR
L a ac
L
( )a
• Generally, as a flux rope expands,
• Problem: Assume that Show that as a flux rope expands, its
magnetic energy decreases as
SOLAR FLUX ROPES: INDUCTANCE
2
2 8ln ,
f
R RL
ac
a fa a
2 2
2
1 8ln
2p t p tf
R RU LI U I
ac
.p tcLI const
1
ln(8 / )pf
UR R a
• Dynamically, the most important interaction is momentum coupling (i.e., forces)
– Drag: Flow around flux rope is not laminar (high magnetic Reynolds’ number)
– Gravity:
SOLAR FLUX ROPES: INTERACTION WITH CORONA
( ) | |d d a i c cF c n m a V V V V
2 ( )( )g i a T
T cav p
F a m g Z n n
n n n
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)
2 22
2 2 2
8 1 1ln 2 1
2 2 2c it t
p g dpp
I Bd Z R R BM F F
a a Bdt c R B Rf
EQUATIONS OF MOTION
t pJ B p p tJ B cJ B
p
fS
2 22
2 2 2
8 1 1ln 2 1
2 2 2c it t
p g dpp
I Bd Z R R BM F F
a a Bdt c R B
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:
EQUATIONS OF MOTION
2 2 22
2 2 2pat t
ppa
I B Bd aM
dt c a B
p tJ B
p
t pJ B p
fS
Fa
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)
Major Radial Equation of Motion:
2 22
2 2 2
8 1 1ln 2 1
2 2 2c it t
p g dpp
I Bd Z R R BM F F
a a Bdt c R B Rf
CHARACTERISTICS OF FLUX-ROPE DYNAMICS
p
fS
,p tcLI
84 ln 2
f
RL R
a
2 222
2 2 22,
ln 8 /
p pR R Rt
f
d ZI f f f
dt L R R a
Major Radial Equation of Motion:
2 22
2 2 2
8 1 1ln 2 1
2 2 2c it t
p g dpp
I Bd Z R R BM F F
a a Bdt c R B Rf
CHARACTERISTICS OF FLUX-ROPE DYNAMICS
p
fS
,p tcLI
84 ln 2
f
RL R
a
2 2
2 22
2
2 2 ln 8 /,p p
R
f
R Rtd Z
dt R RI
af f f
L
• 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)
C2 C2 C2
MAIN ACCELERATION PHASE OF CMEs
Calculated forces
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
222 2
2 2
1 1,
[ ln (8 / )]t
R p Rf
Id Zf f
R Rdt R R a
PHYSICS OF THE MAIN ACCELERATION PHASE
• Basic length scale of the equation of motion
2 1 3,R R R
fs
* 2fS
Z Z
fS
Z Z
*Z
CHARACTERISTIC HEIGHTS
•
• Zm: For Z > Z*, R monotonically increases
• The actual height Zmax of maximum acceleration
22
2 2
1
ln 8 /p R
f
d Zf
d t R R a
22 2
2 2
1
ln(8 / )f peak
d Z d Z
R R ad t d t
21
s. t. (e.g., 1.5 )4m m fZ Z S
* max mZ Z Z
EXAMPLES OF OBSERVED CME EVENTS
• Example
2fR
S / 4Z R
• “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)
*3 3 / 4mZ Z R
HEIGHT SCALES OF MAIN PHASE: Observation
• Test the theory against 1998 June 2 CME: is required
1.5 2f sS R
mz*z
* 3.5 , 6mZ R Z R
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
DATA ANALYSIS: DETERMINATION OF LENGTHS
C2 data
Nobeyama Radioheliograph data
Reverse video
Sf = 0.84 RS
Zmax = 0.55 RS
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 )
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 )
*Z
mZ
Chen et al., ApJ (2006)
EP data
CME data
• A quantitative test of the
accelerating force and
the geometrical assumptions
• A quantitative challenge to
all CME models
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?