energy & helicity flux into the corona
DESCRIPTION
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… - PowerPoint PPT PresentationTRANSCRIPT
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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!
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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
(*)(*)
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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” (*)
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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! (*)
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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
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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
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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
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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
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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.
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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!
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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)!
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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
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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.
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Emerging Omega Loop
(BACK)
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“Trombone Slide” Emergence
(BACK)
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Free Energy from Emergence of Twisted Flux
(BACK)
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Free Energy from Shearing Flows
(BACK)
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Free Energy Propagating Twist
(BACK)
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Free Energy from Non-potential Topology
(BACK)
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Vector Potential Pictorial
(BACK)
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Braiding Motions Inject Helicity
(BACK)
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Vector Potential Pictorial
(BACK)
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Vector Potential Pictorial
(BACK)
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