The Magnetic & Energetic Connection Between the Photosphere & Corona Brian Welsch, Bill Abbett, George Fisher, Yan Li, Jim McTiernan, et al. Why do we care about the corona? Flares & CMEs affect space weather! Why do we care about the photospheric connection to the corona? Therein lies a tale…

Flares & CMEs are powered by the coronal magnetic field, B c. Coronal thermal energy density: n k B T ~ (10 9 ) ( )(10 6 ) erg /cm 3 ~ erg /cm 3 Gravitational energy density in the corona: n m p g  h ~ (10 9 ) ( ) (10 4 ) (10 9 ) erg /cm 3 ~ erg /cm 3 Coronal magnetic energy density: (B c ) 2 /8  ~ (10 2 ) / 10 erg /cm 3 ~ 10 erg /cm 3 L 3 ~ (100 Mm) 3 ~ cm 3  Energy ~ erg (This also means that B c is basically Lorentz force-free: J c x B c = 0.)

The coronal vector magnetic field, B c, however, is not well constrained by observations. Line-of-sight- (LOS)-integrated, LOS-component of B c is measurable in IR –IR polarimetry can also yield direction of B trans, too –e.g., Lin et al. 2004: 20” spatial & 70 min. temporal res. Radio measurements can determine coronal field strength B, under assumptions (n, T, h, B c l ) (e.g., Brosius & White 2006) Coronal EUV & SXR loops determine B c ’s connectivity and direction in plane of sky.

We can measure the photospheric field, B p. How might we relate B p to the evolution of B c ? 1. Statistically relate B p with flares/CMEs. 2. Extrapolate B c from B p. 3. Use  t B p to infer changes in B c. 4. Use  t B p to drive dynamic models of  t B c.

1. Several researchers have related photo- spheric B(x,y; t 0 ) to flares & CMEs in t 0 +/-  t. Strong LOS photospheric fields along PILs (polarity inversion lines) are associated with… –CMEs: Falconer et al. (2004, 2006) –Flares: Schrijver (2007) From Schrijver (2007)

1. Several researchers have related photo- spheric B(x,y; t 0 ) to flares & CMEs in t 0 +/-  t. Discriminant Function Analysis (DFA) applied to properties of B, J z, etc. –Leka & Barnes (2003a,b; 2006 [v.v]; 2007) –2007 study: “we conclude that the state of the photospheric magnetic field at any given time has limited bearing on whether that region will be flare productive.” –Generally, the best bet is: The corona will not flare.

2. One can extrapolate 3D coronal B c (x,y,z; t 0 ) from 2D photospheric B p (x,y; t 0 ). PFSS extrapolations assume J = 0. (Not true!) Useful as initial condition for MHD models. –Step 2: “turn on” the solar wind… ASIDE: Li & Luhmann (2006) studied model pre- CME fields, and found that most were dipolar. DIPOLAR - inconsistent with “breakout” QUADRUPOLAR

2. One can extrapolate 3D coronal B c (x,y,z; t 0 ) from 2D photospheric B p (x,y; t 0 ). Non-linear force-free (NLFF) extrapolations assume J || B c. –e.g., Schrijver et al. (2005), analytic test B –e.g., Metcalf et al. (2006), data-inspired test B –e.g., Schrijver et al. (2008), Hinode B p

When applied to Hinode data, uncertainties in energies of extrapolated fields are large.

3. Relate evolution of photospheric B p to evolution of coronal B c. Faraday’s Law relates magnetic evolution to the electric field:  t B = -c (  x E). The photospheric electric field E p controls the flux of magnetic energy & helicity into B c : dU/dt = c ∫ dA [ z ∙ (E p x B p )] /4π dH/dt = 2 ∫ dA [ z ∙ (E p x A p )] ^ ^

3. Relate evolution of photospheric B p to evolution of coronal B c. Both (  x E p ) and (  ∙ E p ) determine E p ; but  t B p only specifies the former, “inductive” component, E p I. –E p is known to the gradient of a scalar,  E p = -   –This is a “gauge” problem for E (cf., gauge freedom on A) Ideal evolution corresponds to E p B p = 0. Since (  x E p ) depends on  t B p, not B p itself, ideality is not automatic. –Hence, one can impose  to force E p B p = 0. See George Fisher’s poster, 01-04, for more about finding E p I and  !

3. Relate evolution of photospheric B p to evolution of coronal B c. Imposing E p B p = 0 and (  x E p ), however, still does not fully constrain E p ! –Call an inductive & ideal electric field E II. –One can add the gradient of another scalar,  E p = - , assuming this scalar is also perpendicular to B p. Since ideality means E p = -(v x B p )/c,  E p corresponds to a flow,  v. –What are the properties of  v? How can  v be found?

Q: How do we determine  v? A: Tracking! Here’s an example of local correlation tracking (LCT) flows.

The tracked apparent motion of magnetic flux in magnetograms is flux transport velocity, u f. u f is not equivalent to v; rather, u f  v h - (v z /B z )B h u f is the apparent velocity (2 components) v is the actual plasma velocity (3 comps),  to B p (NB: non-ideal effects can also cause flux transport!) Démoulin & Berger (2003): In addition to horizontal flows, vertical velocities can lead to u f  0. In this figure, v h = 0, but v z  0, so u f  0. (NB: notation in fig. differs)

Flows inferred from tracking can constrain  E. E h = z x u f B z = E h II -  h    h x  z = u f B z - E h II x z   h 2  =  h x ( u f B z - E h II x z) Ideality then implies that E z = - (E h B h )/B z These two conditions insure that  won’t affect  t B z. To insure that  won’t affect  t B h, one must also assume  z E h =  h E z. ^ ^ ^ ^

3. (still!) Relate evolution of photo- spheric B p to evolution of coronal B c.  t B + tracking can be used to estimate E p Welsch et al. (2007) tested tracking methods with flows from MHD simulations. – estimates of energy & helicity fluxes were poor! – even the best methods confounded shearing & emergence There is much room for improvement!

3. Relate evolution of photospheric B p to evolution of coronal B c. In its own right, E p can be used to understand evolution of the coronal magnetic field B c. But E p is also useful for other types of modeling…

4. Dynamic models of coronal B c (x,y,z; t) can be driven from time series of photospheric B p (x,y; t). One can be evolve B c (x,y,z; t i ) via the induction equation alone. - e.g., Yeates, Mackay, & Van Ballegooijen (2008) This approach cannot accurately model rapid coronal evolution.

4. Dynamic models of coronal B c (x,y,z; t) can be driven from time series of photospheric B p (x,y; t). The many scale heights in (n,T) between the photosphere & corona makes incorporation of B p into coronal models challenging. RADMHD (Abbett 2007) was developed to overcome this. RADMHD can include convection in AR-scale magnetic field models, as shown in this preliminary run.

4. Dynamic models of coronal B c (x,y,z; t) can be driven from time series of photospheric B p (x,y; t). RADMHD includes empirical parame- terizations of radiation, as well as thermal cond- uction. RADMHD can include convection in AR-scale magnetic field models, as shown in this preliminary run.

4. Dynamic models of coronal B c (x,y,z; t) can be driven from time series of photospheric B p (x,y; t). RADMHD also can include effects of convective flows in the bottom layers of coronal models. Note effect of convective cells on orientations of magnetic vector orientations in this partially relaxed configuration.

How is the vector magnetic field determined? Magnetic fields will be split by Zeeman effect, but using the split itself not useful in most cases. Spot in 5250 A (normal Zeeman triplet) Slide courtesy of Tom Metcalf, CoRA/NWRA

Observed Stokes Profiles Na-D line observations from the IVM They look more or less as expected with a few differences: –Noise is clearly present –prefilter distorts spectrum Stokes IStokes QStokes UStokes V Relative Wavelength (nm) Slide courtesy of Tom Metcalf, CoRA/NWRA

3. Inverting the Polarization Observations to get B The best method is to observe the Zeeman splitting directly. –Not generally possible for optical observations since the fields on the Sun are too weak. –The Zeeman splitting goes as so this works better in the IR. –Gives the magnetic field directly without worrying about the filling factor. The next best method is to fit the Stokes profiles to the Unno profiles (Milne-Eddington atmosphere: source function linear with optical depth). –This gives the magnetic field, filling factor, thermodynamic parameters Slide courtesy of Tom Metcalf, CoRA/NWRA

4. Dynamic models of coronal B c (x,y,z; t) can be driven from time series of photospheric B p (x,y; t). Inverting polarimetric observations for B p requires estimating thermodynamic variables (e.g., T & v). Q: When B p is then used in a dynamic model, how do the “inversion” & model’s (T, v) compare? Should models driven by polarimetric data be used in the inversion process to estimate B p ?

I have reviewed several ways researchers are attempting to relate B p to B c. 1. Statistically relate B p with flares/CMEs. 2. Extrapolate B c from B p. 3. Use  t B p to infer changes in B c. 4. Use  t B p to drive dynamic models of  t B c. I noted problems with #1, #2, and #3. I suspect that #4 is simply too new to have found problems yet.

Conclusions We still cannot reliably use B p to deter- mine when B c is flare- or CME-prone. It’s not for lack of good ideas – some of which are still in their being developed. But surely there are more good ideas that remain to be tried! Your thoughts?