The Magnetic & Energetic Connection Between the Photosphere & Corona Brian Welsch, Bill Abbett, George Fisher, Yan Li, Jim McTiernan, et al. Why do we.

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, Bc.

• Coronal thermal energy density: n kB T ~ (109) (10-16)(106 ) erg /cm3

~ 10-1 erg /cm3

• Gravitational energy density in the corona:n mp g h ~ (109) (10-24) (104 ) (109 ) erg /cm3

~ 10-2 erg /cm3

• Coronal magnetic energy density: (Bc)2/8 ~ (102) / 10 erg /cm3

~ 10 erg /cm3

L3 ~ (100 Mm)3 ~ 1030 cm3 Energy ~ 1031 erg

(This also means that Bc is basically Lorentz force-free: Jc x Bc = 0.)

The coronal vector magnetic field, Bc, however,

is not well constrained by observations.

• Line-of-sight- (LOS)-integrated, LOS-component of Bc is measurable in IR

– IR polarimetry can also yield direction of Btrans, 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, Bc • l)

(e.g., Brosius & White 2006)

• Coronal EUV & SXR loops determine Bc’s connectivity and direction in plane of sky.

We can measure the photospheric field, Bp. How might we relate Bp to the evolution of Bc?

1. Statistically relate Bp with flares/CMEs.

2. Extrapolate Bc from Bp.

3. Use tBp to infer changes in Bc.

4. Use tBp to drive dynamic models of tBc.

1. Several researchers have related photo-spheric B(x,y; t0) to flares & CMEs in t0 +/- 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; t0) to flares & CMEs in t0 +/- t.

• Discriminant Function Analysis (DFA) applied to properties of B, Jz, 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 Bc(x,y,z; t0) from 2D photospheric Bp(x,y; t0).

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 Bc(x,y,z; t0) from 2D photospheric Bp(x,y; t0).

• Non-linear force-free (NLFF) extrapolations assume J || Bc.

– 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 Bp

When applied to Hinode data, uncertainties

in energies of extrapolated fields are large.

3. Relate evolution of photospheric Bp to evolution of coronal Bc.

• Faraday’s Law relates magnetic evolution to the electric field: tB = -c ( x E).

• The photospheric electric field Ep controls the flux of magnetic energy & helicity into Bc:

dU/dt = c ∫ dA [ z ∙ (Ep x Bp)] /4π

dH/dt = 2 ∫ dA [ z ∙ (Ep x Ap)]

^

^

3. Relate evolution of photospheric Bp to evolution of coronal Bc.

• Both ( x Ep) and ( ∙ Ep) determine Ep; but tBp only specifies the former, “inductive” component, Ep

I.

– Ep is known to the gradient of a scalar, Ep = -– This is a “gauge” problem for E (cf., gauge freedom on A)

• Ideal evolution corresponds to Ep • Bp = 0. Since ( x Ep) depends on tBp, not Bp itself, ideality is not automatic.

– Hence, one can impose to force Ep • Bp = 0. See George Fisher’s poster, 01-04, for more about finding Ep

I and !

3. Relate evolution of photospheric Bp to evolution of coronal Bc.

• Imposing Ep • Bp = 0 and ( x Ep), however, still does not fully constrain Ep!

– Call an inductive & ideal electric field EII.

– One can add the gradient of another scalar, Ep = - , assuming this scalar is also perpendicular to Bp.

• Since ideality means Ep = -(v x Bp)/c, Ep 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, uf.

uf is not equivalent to v; rather, uf vh - (vz/Bz)Bh

• uf is the apparent velocity (2 components)

• v is the actual plasma velocity (3 comps), to Bp

(NB: non-ideal effects can also cause flux transport!)

Démoulin & Berger (2003): In addition to horizontal flows, vertical velocities can lead to uf 0. In this figure, vh= 0, but vz 0, so uf 0.

(NB: notation in fig. differs)

Flows inferred from tracking can constrain E.

• Eh = z x ufBz = EhII -h h x z = ufBz - Eh

II x z

h2 = h x ( ufBz - Eh

II x z)

• Ideality then implies that Ez = - (Eh • Bh)/Bz

• These two conditions insure that won’t affect tBz.

• To insure that won’t affect tBh, one must also assume zEh = hEz .

^ ^^

^

3. (still!) Relate evolution of photo-spheric Bp to evolution of coronal Bc.

tB + tracking can be used to estimate Ep

• 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 Bp to evolution of coronal Bc.

• In its own right, Ep can be used to understand evolution of the coronal magnetic field Bc.

• But Ep is also useful for other types of modeling…

4. Dynamic models of coronal Bc(x,y,z; t) can be driven from time series of photospheric Bp(x,y; t).

One can be evolve Bc(x,y,z; ti) 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 Bc(x,y,z; t) can be driven from time series of photospheric Bp(x,y; t).

The many scale heights in (n,T) between the photosphere & corona makes incorporation of Bp into coronal models challenging.

RADMHD (Abbett2007) 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 Bc(x,y,z; t) can be driven from time series of photospheric Bp(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 Bc(x,y,z; t) can be driven from time series of photospheric Bp(x,y; t).

RADMHD also caninclude effects ofconvective flows in the bottom layersof coronal models.

Note effect of convective cells on orientations of magnetic vector orientations in this partially relaxed configuration.

4. Dynamic models of coronal Bc(x,y,z; t) can be driven from time series of photospheric Bp(x,y; t).

Inverting polarimetric observations for Bp requires estimating thermodynamic variables (e.g., T & v).

Q: When Bp 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 Bp?

I have reviewed several ways researchers are attempting to relate Bp to Bc.

1. Statistically relate Bp with flares/CMEs.

2. Extrapolate Bc from Bp.

3. Use tBp to infer changes in Bc.

4. Use tBp to drive dynamic models of tBc.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 Bp to deter-mine when Bc 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?