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

A FLUX-ROPE CME

CME leading edge (ZLE)

EP (Zp)

Chen (JGR, 1996)

LASCO C2 data: 12 Sept 2000

x

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

Example

• 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?