Energy & Helicity Flux into the Corona

• I’ll discuss secular trends, ‘the relatively consistent movement of a variable over a long period’ – ‘DC’ as opposed to ‘AC’ (wave) flux

• I’ll assume ideal MHD applies, E = -(v x B)/c

• First, express fluxes in equations…

• …then in terms of boundary motions that

we all know and love!

Energy Flux

• Poynting Flux, with E = -(v x B)/c:

• Rewrite using BAC – CAB rule, take z-comp:

B)/4 x B) x ((v - B)/4 x c(ES

4 / ]

shearing

) (B -

emergence

) ( v[

4 / ] )(B - )( [v

4 / z ] )()([ S

hhzhhz

zz

z

BvBB

B vB B

B v B - B B v

(*)(*)

Emergence of Energy

• Archetype is vertical, perpendicular flow – e.g., vz advecting a flux tube’s apex (*)

• NB: does not include vertical, parallel flow – e.g., continued emergence of a “U-bolt” (*)

Emergence of Free Energy (EF)

EF from emerging non-potential fields (*) related to helicity emergence

EF from shearing or twisting (*) propagation of twist (*)(cf., Nightingale’s rotating sunspots)

• possibly from emerging potential fields into potential background fields!

Depends on global topology! (*)

Relative Helicity, Defined

• Refers to two B-fields and their vector potentials:1. B is the actual magnetic field, which may or may not vanish (Bn

0) on the boundary V of V.

2. BP is the current-free magnetic field that matches the actual field Bn on the boundary.

• A is a vector pot’l for B, B = ( x A), and

AP is vector pot’l for BP (“potential field’s potential”).

• In the general case (barring symmetry), B BP H 0.

• This helicity is gauge invariant (Berger & Field, 1984).

)B - (B )A (A dVH PP

Vector Potential, AP

• By construction, n · ( x A) = Bz ,

( ·AP)= 0, and (AP· n) = 0.

• Regions of Bz 0 on the photosphere are sources of AP, which is a 2-D vector field.

• AP is circumferential (*), and completely specified by Bz – knowing Bh unnecessary!

)'- ( x z '-

)'(B 'Sd

2

1)(A 2

z2P xx

xx

xx

Helicity Flux – Poynting-like Eqn.

• From Faraday’s Law for B/t , and assuming A/t = -cE = (v x B), Berger & Field computed dH/dt across a surface (z = normal direction)

• Rz is a helicity flux density (Mx2/cm2/s).

]

shearing

) (B -

emergence

) ( v[ 2 R hPzhPzz vABA

Emergence of Helicity

1. Emergence of helicity-carrying (“helical”—

basically, twisted!) fields. (*)– e.g., current-carrying fields observed

by Leka et al. (1996)

2. Emergence of non-helical (potential) fields, resulting in non-potential topology!

Depends on global topology! (*)

)B - (B )A (A H PP

Helicity Flux by Shearing/Twisting

• Motion of magnetic flux along contours of AP leads to a flux of helicity. (*)

– Contours are circular, so relative circular motion, a.k.a., braiding, leads to helicity injection.

– Propagation of twist along flux tube (*) should lead to observable foot point braiding.

Q. How can we determine the photospheric velocity?

• (Not just any photospheric velocity – we want the velocity that affects the photospheric magnetic field.)

• Doppler info can give LOS velocity, e.g., Chae (2004)

• Local Correlation Tracking (LCT), e.g., Chae (2001)

• Newer methods: Kusano et al. (2002), ILCT, MEF, Minimum Structure, ??? – More at Velocity Shootout in WG1 splinter session on Fri!

Démoulin & Berger’s (2003) Hypothesis

• LCT applied to Bz or BLOS from photospheric magnetograms gives the pattern velocity.

• Pattern velocity uLCT is related to plasma velocity v by:

hzzhLCT )Bv( Bvu

hzzhzLCT vBB Bvu dE/dt and dH/dt can be written in terms of (uLCT Bz)!

dE/dt and dH/dt in terms of uLCT Bz

• From LCT on vector magnetograms, can compute fluxes of magnetic energy and helicity.

• With LOS magnetograms, and an assumed Bh (e.g., FFF that matches coronal observations), can estimate energy & helicity fluxes.

4/ )(B

4 / )] (B -)( v[ R

4/ )(B

4 / )] (B -) ( v[ S

PLCTz

PhzPhzz

hLCTz

hhzhhzz

Au

AvAB

Bu

BvBB

Helicity & Energy Fluxes

• Barring symmetry, if B BP, then H 0.

|dH/dt| > 0 B BP

• Since BP is unique, and minimum energy given Bz(x,y), B BP also means free energy is present, EF 0.

• Knowledge of Bz specifies AP.

• Knowledge of v allows computation of dE/dt & dH/dt.

Emerging Omega Loop

(BACK)

“Trombone Slide” Emergence

(BACK)

Free Energy from Emergence of Twisted Flux

(BACK)

Free Energy from Shearing Flows

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Free Energy Propagating Twist

(BACK)

Free Energy from Non-potential Topology

(BACK)

Vector Potential Pictorial

(BACK)

Braiding Motions Inject Helicity

(BACK)

Vector Potential Pictorial

(BACK)

Vector Potential Pictorial

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