Flux Emergence & the Storage of Magnetic Free Energy Brian T. Welsch Space Sciences Lab, UC-Berkeley Flares and coronal mass ejections (CMEs) are driven by the release of free magnetic energy stored in the coronal magnetic field. But fundamental questions about this “storage and release” paradigm remain open, such as: How does free magnetic energy enter the corona? How is this energy stored? What triggers its release? How can ground-based solar observations address these questions?

Flares and CMEs are powered by energy in the coronal magnetic field. From T.G. Forbes, “A Review on the Genesis of Coronal Mass Ejections”, JGR (2000)

The hypothetical coronal magnetic field with lowest energy is current-free, or “potential.” For a given coronal field B cor, the coronal magnetic energy is: U   dV (B cor · B cor )/8 . The lowest energy coronal field would have current J = 0, and Ampére says 4 π J/c =  x B, so  x B min = 0. A curl-free vector field can be expressed as the gradient of a scalar potential, B min = - . (Since  ·B min = -  2  = 0, it’s easy to solve!) U min   dV (B min · B min )/8  The difference U (F) = [U – U min ] is “free” energy stored in the corona, which can be suddenly released in flares or CMEs.

Coronal J cannot currently be observationally constrained; measurements of (vector) B cor (x, y, z) haven’t been made. In consequence, we must use indirect means to infer properties of coronal currents. When not flaring, B photosphere and B chromosphere are coupled to B cor, so provide valuable insights. While B cor can evolve on its own, changes in B ph or B ch will induce changes in the coronal field B cor. In addition, following active region (AR) fields in time can provide information about their history and development.

Unfortunately, our ignorance regarding free magnetic energy in the corona is profound! 1. Physically, how does free energy enter the corona? – Practically, how can we detect this buildup? 2. Physically, how is this energy stored? – Practically, how can we quantify it once it’s there? 3. Physically, what triggers its release? – Practically, how can we predict when release is imminent? Common theme: we want to know vector B (so we can infer J) from the photosphere upward into the corona.

Some Key Roles for Ground Based Efforts: Develop means to estimate vector B(x,y,z) in chromosphere and corona (radio, IR).

Short answer to #1: Energy comes from the interior! But how? Image credits: George Fisher, LMSAL/TRACE EUV image of ~1MK plasma

What physical processes produce the electric currents that store energy in B cor ? Two options are: (i) Currents could form in the interior, then emerge into the corona. – Current-carrying magnetic fields have been observed to emerge (e.g., Leka et al. 1996, Okamoto et al. 2008). (ii) Photospheric evolution could induce currents in already- emerged coronal magnetic fields. – From simple scalings, McClymont & Fisher (1989) argued induced currents would be too weak to power large flares. – Longcope et al. (2007), Kazachenko et al. (2009), and others argue that strong enough currents can be induced. Both models involve slow buildup, then sudden release.

For (i), note that currents can emerge in two distinct ways! (Should be twisted, but I can’t draw twist very well!) a) emergence of new flux b) vertical transport of cur- rents in emerged flux NB: This does not increase total unsigned photospheric flux. Ishii et al., ApJ v.499, p NB: New flux only emerges along polarity inversion lines!

For (ii), if coronal currents induced by post-emergence photospheric evolution drive flares and CMEs, then: The evolving coronal magnetic field must be modeled! NB: Induced currents close along or above the photosphere --- they are not driven from below. ==> All available energy in these currents can be released. Longcope, Sol. Phys. v.169, p

Back to the catalog of our ignorance regarding free magnetic energy in the corona: 1. Physically, how does free energy enter the corona? – Practically, how can we detect this buildup? 2. Physically, how is this energy stored? – Practically, how can we quantify it once it’s there? 3. Physically, what triggers its release? – Practically, how can we predict when release is imminent?

Back to the catalog of our ignorance regarding free magnetic energy in the corona: 1. Physically, how does free energy enter the corona? – Practically, how can we detect this buildup? Statistically? 2. Physically, how is this energy stored? – Practically, how can we quantify it once it’s there? 3. Physically, what triggers its release? – Practically, how can we predict when release is imminent?

Statistical methods have been used to correlate observables with flare & CME activity, including: Total flux in active regions, vertical current (e.g., Leka et al. 2007) Flux near polarity inversion lines (PILs; e.g., Falconer et al ; Schrijver 2007) “Proxy” Poynting flux, v h B R 2 (e.g., Welsch et al. 2009) Subsurface flows (e.g., Reinard et al. 2010, Komm et al. 2011) Magnetic power spectra (e.g., Abramenko & Yurchyshyn, 2010) It’s challenging to infer physics from correlations, so I will emphasize more deterministic approaches here.

Back to the catalog of our ignorance regarding free magnetic energy in the corona: 1. Physically, how does free energy enter the corona? – Practically, how can we detect this buildup? Catch it in the act! 2. Physically, how is this energy stored? – Practically, how can we quantify it once it’s there? 3. Physically, what triggers its release? – Practically, how can we predict when release is imminent?

In principle, electric fields derived from magnetogram evol- ution can quantify energy or helicity fluxes into the corona. The fluxes of magnetic energy & helicity across the magnetogram surface into the corona depend upon E: dU/dt = ∫ dA (E x B) z /4 π dH/dt = 2 ∫ dA (E x A) z U and H probably play central roles in flares / CMEs. Coupling of B cor to B beneath the corona implies estimates of E there provide boundary conditions for data-driven, time-dependent simulations of B cor.

One can use either  t B z or, better,  t B to estimate E or v. “Component methods” derive v or E h from the normal component of the ideal induction equation,  B z /  t = -c[  h x E h ] z = [  x (v x B) ] z But the vector induction equation can place additional constraints on E:  B/  t = -c(  x E)=  x (v x B), where I assume the ideal Ohm’s Law,* so v E: E = -(v x B)/c ==> E·B = 0 *One can instead use E = -(v x B)/c + R, if some model resistivity R is assumed. (I assume R might be a function of B or J or ??, but is not a function of E.)

Tracking with “component methods” constrains ψ by estimating u in the source term (  h x u B z ) · z. Methods to find ψ via tracking include, e.g.: – Local Correlation Tracking (LCT, November & Simon 1988; ILCT, Welsch et al. 2004; FLCT Fisher & Welsch 2008) – the Differential Affine Velocity Estimator (DAVE, and DAVE4VM; Schuck 2006 & Schuck 2008) (Methods to find ψ via integral constraints also exist, e.g., Longcope’s [2004] Minimum Energy Fit [MEF] method.) Welsch et al. (2007) tested some of these methods using “data” from MHD simulations; MEF performed best. Further tests with more realistic data are underway. ^

While  t B provides more information about E than  t B z alone, it still does not fully determine E. Faraday’s Law only relates  t B to the curl of E, not E itself; a gauge electric field  ψ is unconstrained by  t B. (Ohm’s Law does not fully constrain E.)  t B h also depends upon vertical derivatives in E h, which single-height magnetograms do not fully constrain. Doppler data can provide additional info.

The ideal induction equation is:  t B x = (-  y E z +  z E y )c =  y (v x B y - v y B x ) -  z (v z B x - v x B y )  t B y = (-  z E x +  x E z )c =  z (v y B z - v z B y ) -  x (v x B y - v y B x )  t B z = (-  x E y +  y E x )c =  x (v z B x - v x B z ) -  y (v y B z - v z B y )

B h (t i ) v B h (t f ) v B h (t i )  t B h =  z (E h x z) - (  h x E z z) vertical shear in v hor ^ converging/diverging surface flows Horizontal flows with either vertical shear or nonzero horizontal divergence (or both) alter the horizontal field B h. ^ If only vertical shear causes  t B h, then v = 0 at the photosphere, so no evident vertical Poynting flux!  z E h estimated from magnetograms at different heights (e.g., HMI + SOLIS) can constrain which process is at work.

There are several open questions regarding electric fields in multi-height magnetogram sequences. One can estimate E in sequences magnetograms in spectral lines with different formation heights – chromosph. LOS & vector photosph., SOLIS 854.2nm + HMI – (also worthwhile to baseline SOLIS & HMI photospheric) Compare electric fields / flows. – Where are they correlated in space? Where are they not? (e.g., in strong field vs. weak field regions?) – Do flow vorticities / divergences appear in common loci? – Do flows decorrelate on similar timescales? Inversions of the chromospheric vector field would certainly be useful, but also certainly not necessary.

Some Key Roles for Ground Based Efforts: Develop means to estimate vector B(x,y,z) in chromosphere and corona (radio, IR). Routine magnetograms from atmospheric layers different than space-based observations. (Cadence c. 30 min?)

Autocorrelation of LCT flows suggest 96-min. cadence for 2’’ MDI magnetograms is not unreasonably slow. BLACK shows autocorrelation for B R ; thick is current-to-previous, thin is current-to-initial. BLUE shows autocorrelation for u x ; thick is current-to-previous, thin is current-to-initial. RED shows autocorrelation for u y ; thick is current-to-previous, thin is current-to-initial.

Autocorrelation of LCT flows suggest 48-min. cadence for HMI magnetograms binned 2x2 (so 1’’ pix) is not bad. Upper, nearly flat lines are correlations between successive x- and y- flow maps, with 48-min. cadence. Decreasing lines are correlations between i-th and initial flow maps.

Autocorrelation of B LOS shows persistence of magnetic structure. Hinode Stokes V binned 1x1, 2x2, 4x4, … 64x64

While  t B provides more information about E than  t B z alone, it still does not fully determine E. Faraday’s Law only relates  t B to the curl of E, not E itself; a gauge electric field  ψ is unconstrained by  t B. (Ohm’s Law does not fully constrain E.)  t B h also depends upon vertical derivatives in E h, which single-height magnetograms do not fully constrain. Doppler data can provide additional info.

Doppler data helps because emerging flux might have little or no inductive signature at the emergence site. Schematic illustration of flux emergence in a bipolar magnetic region, viewed in cross-section normal to the polarity inversion line (PIL). Note the strong signature of the field change at the edges of the region, while the field change at the PIL is zero.

Aside: Flows v || along B do not contribute to E = -(v x B)/c, but do “contaminate” Doppler measurements. v LOS v v v =

Aside: Dopplergrams are sometimes consistent with “siphon flows” moving along B. MDI Dopplergram at 19:12 UT on 2003 October 29 superposed with the magnetic polarity inversion line. (From Deng et al. 2006) Why should a polarity inversion line (PIL) also be a velocity inversion line (VIL)? One plausible explanation is siphon flows arching over (or ducking under) the PIL. What’s the DC Doppler shift along this PIL? Is flux emerging or submerging?

For instance, the “PTD” method (Fisher et al. 2010, 2011) can be used to estimate E: In addition to  t B z, PTD uses information from  t J z in the derivation of E. No tracking is used to derive E, but tracking methods (ILCT, DAVE4VM) can provide extra info! Using Doppler data improves PTD’s accuracy! For more about PTD, see Fisher et al (ApJ ) and Fisher et al (Sol. Phys. in press; arXiv: ).

The “PTD” method employs a poloidal-toroidal decomposition of B into two scalar potentials. B =  x (  x B z) +  x J z B z = -  h 2 B, 4 π J z /c =  h 2 J,  h ·B h =  h 2 (  z B ) Left: the full vector field B in AR Right: the part of B h due only to J z. ^^  t B =  x (  x  t B z) +  x  t J z  t B z =  h 2 (  t B ) 4 π  t J z /c =  h 2 (  t J )  h ·(  t B h ) =  h 2 (  z (  t B )) ^^

How accurate is PTD? We tested it with MHD simulations of emerging flux also used by Welsch et al. (2007). Top row: Three components of E and S z from MHD code. 2nd row: PTD E and S z 3rd row: PTD + Doppler E and S z. (Note the dramatic improvement in the estimate of S z !) 4th row: PTD + FLCT E and S z. 5th row: PTD + Doppler + FLCT E and S z (Note good recovery of E x, E y, and S z, and reduction in artifacts in weak-E regions.)

Quantitative tests with simulated data show Doppler information improves recovery of E-field and Poynting flux S z. Upper right: MHD S z vs. PTD + Doppler S z. Lower right: MHD S z vs. PTD + Doppler + FLCT S z. Poynting flux units are in [10 5 G 2 km s −1 ] Upper left: MHD S z vs. PTD S z. Lower left: MHD S z vs. PTD + FLCT S z.

Problem: With real observations, convective blueshifts must be removed! There’s a well-known intensity-blueshift correlation, because rising plasma (which is hotter) is brighter (see, e.g., Gray 2009; Hamilton and Lester 1999; or talk to P. Scherrer). (Helioseismology uses time differences in Doppler shifts, so this issue isn’t a problem.) Because magnetic fields suppress convection, lines are redshifted in magnetized regions. Consequently, absolute calibration of Doppler shifts is essential. From Gray (2009): Bisectors for 13 spectral lines on the Sun are shown on an absolute velocity scale. The dots indicate the lowest point on the bisectors. (The dashed bisector is for λ6256.) Lines formed deeper in the atmosphere, where convective upflows are present, are blue- shifted.

HMI data clearly exhibit this effect. How can this bias best be corrected? An automated method (from Welsch & Li 2008) identified PILs in a subregion of AR Note predominance of redshifts.

Some Key Roles for Ground Based Efforts: Develop means to estimate vector B(x,y,z) in chromosphere and corona (radio, IR). Routine magnetograms from atmospheric layers different than space-based observations. Cadence c. 30 min.? Minor: Estimates of absolute Doppler shifts.

Back to the catalog of our ignorance regarding free magnetic energy in the corona: 1. Physically, how does free energy enter the corona? – Practically, how can we detect this buildup? Catch it in the act! 2. Physically, how is this energy stored? – Practically, how can we quantify it once it’s there? Can infer existence of energy from coronal observations… 3. Physically, what triggers its release? – Practically, how can we predict when release is imminent?

A cartoon model helps bias the mind… A quasi-stable balance can exist between outward magnetic pressure and inward magnetic tension. From Moore et al. (2001)

Non-potential fields are evinced by filaments / prominences, and sheared H- α fibrils & coronal loops. Non-potential structures can persist for weeks, then flare or erupt suddenly. Courtesy Tom Berger

Back to the catalog of our ignorance regarding free magnetic energy in the corona: 1. Physically, how does free energy enter the corona? – Practically, how can we detect this buildup? Catch it in the act! 2. Physically, how is this energy stored? – Practically, how can we quantify it once it’s there? Requires quantitative modeling of coronal field. 3. Physically, what triggers its release? – Practically, how can we predict when release is imminent?

The Minimum Current Corona (MCC) approach can be used to estimate a lower-bound on coronal energy. Method: Determine linkages from initial magnetogram, infer coronal currents (and therefore energy) based upon magnetogram evolution. Separators with large currents have been related to flare sites. Kazachenko et al. (2011)

Mark Cheung has been running magnetogram-driven coronal models.

The magnetofrictional approach has a number of shortcomings.

Accurate driving of the model requires accurate estimation of the boundary electric field E.

The assumption that  h ·E h = 0 results in little free energy.

Mark gets more free energy with an ad-hoc assumption for  h ·E h -- estimates of E from observations would be better!

Some Key Roles for Ground Based Efforts: Develop means to estimate vector B(x,y,z) in chromosphere and corona (radio, IR). Routine magnetograms from atmospheric layers different than space-based observations. Cadence c. 30 min.? Minor: Estimates of absolute Doppler shifts. Modeling!

Back to the catalog of our ignorance regarding free magnetic energy in the corona: 1. Physically, how does free energy enter the corona? – Practically, how can we detect this buildup? Catch it in the act! 2. Physically, how is this energy stored? – Practically, how can we quantify it once it’s there? Requires quantitative modeling of coronal field. 3. Physically, what triggers its release? – Practically, how can we predict when release is imminent? Again, quantitative modeling of the coronal field is needed.

“Potentiality” should imply little free energy, and little likelihood of flaring. Schrijver et al. (2005) found potential ARs were relatively unlikely to flare.

“Non-potentiality” should imply non-zero free energy, and increased likelihood of flaring. Schrijver et al. (2005) found non-potential ARs were more likely to flare, with fields becoming more potential over hours.

Non-potential structures can persist for weeks, then flare or erupt suddenly. Hudson et al. (1999) The hot, “chewy nougat” in the core of this non-potential structure --- visible in SXT --- persists for months. Evidently, the corona can store free energy for long times! What would cause this to erupt? Inferring the presence of free energy is not enough to predict its release!

How can coronal “susceptibility” to destabilization (from emergence, or twisting, or reconnection) be inferred? Schrijver includes several effects here; he argues only one (emergence) triggers eruptions. But this is speculative! Schrijver ASR v. 43 p

Hard X-ray (HXR) flare emission (RHESSI) should be related to chromospheric magnetic properties. HXR emission is bremsstrahlung from non-thermal particles precipitating onto the upper chromosphere. Magnetic mirroring is thought to govern the precipitation of flare particles. – If so, in magnetic fields that converge more strongly, more particles should mirror, and less HXR emission should be seen. Models of the potential magnetic field at the chromo- sphere (e.g., from SOLIS data) quantify field convergence. HXR intensities & indices of power-law fits to emission spectra can be compared to aspects of chromospheric B.

Some Key Roles for Ground Based Efforts: Develop means to estimate vector B(x,y,z) in chromosphere and corona (radio, IR). Routine magnetograms from atmospheric layers different than space-based observations. Cadence c. 30 min.? Minor: Estimates of absolute Doppler shifts. Modeling! And more modeling!

Recommendations for Ground Based Efforts: Develop means to estimate vector B(x,y,z) in chromosphere and corona (radio, IR).  Facilities: ATST, FASR, COMP/COSMO; modeling! Routine magnetograms from atmospheric layers different than space-based observations. Cadence c. 30 min.?  Global networks: GONG; SOLIS-like vect. m’graphs Modeling! And more modeling!  Support both “practical” (for SWx) & “impractical” (for curiosity) efforts, esp. data-driven modeling

The End