Can We Determine Electric Fields and Poynting Fluxes from Vector Magnetograms and Doppler Shifts?

by George Fisher, Brian Welsch, and Bill AbbettSpace Sciences Lab, UC-Berkeley

See: http://arxiv.org/abs/1101.4086

We can use magnetic evolution observed by HMI to infer the vertical Poynting flux at the photospere.

• The electric field E appears in the vertical Poynting flux of magnetic energy across the photosphere,

• HMI measures ΔB/Δt at the photosphere, and Faraday told us:

The z-comp. of (6), the z-comp. of its curl, and its horizontal divergence yield 3 Poisson equations,

Eqns. (7)-(9) can be solved using observed data to infer the scalar potentials.

See Fisher et al. (2010) for details regarding solution of these equations.

Unfortunately, important magnetodynamics is not always apparent in ΔB/Δt -- e.g., flux emergence!

Figure 1. Schematic illustration of flux emergence in a bipolar active region, viewed in cross-section normal to the polarity inversion line (PIL).

The emerging flux is rising at a speed vz, which could be measured by an observer viewing the active region from above. The length of the bipolar active region (the distance from the edge of one pole to the edge of the other pole) at the time illustrated is 2x0.

Note the strong signature of the field change at the edges of the active region, while the field change at the PIL is zero.

What additional information can be used to constrain the inferred electric field E? Doppler data!

• Along PILs of the line-of-sight (LOS) field, Doppler shifts and transverse magnetic fields unambiguously determine the “Doppler electric field:”

• Away from PILs, flows along B (which are unrelated to E) contribute to Doppler shifts, so we can’t use non-PIL Doppler data

==> Keep PTD solution for EI in non-PIL regions!

How can we combine information from ED near PILs with PTD solutions for EI away from PILs?

• We use small values of |Bz|/|Bh| to define a confidence function w weighted by proximity to PILs ,

where σ is a parameter we set to 0.6 here.• We define the PIL-weighted “modulated E field,” EM,

from which we determine a curl-free electric field Eχ (since the curl of EM is matched by the PTD EI field):

Next, we combine Eχ and the PTD EI field, and then find another potential field to ensure total E is ideal.

• The total magnetic field is given by

where we used the iterative scheme of Fisher et al. (2010) to determine the scalar potential ψ,

so that E.B=0, as implied by E = -(v x B)/c.

Away from PILs, we can also incorporate information from local correlation tracking (LCT).

• We used the FLCT code (Fisher & Welsch 2008) to estimate a horizontal electric field,

which we weight by the complementary (non-PIL) confidence function, and use to determine a curl-free electric field Eζ consistent with ELCT,

and then compute a total, ideal E that is a hybrid of the PTD + Doppler + LCT electric fields.

How accurate are our methods? We tested them with MHD simulations of emerging flux from Welsch et al. (2007).

Figure 2. Top row: The three components of the electric field E and the vertical Poynting flux Sz from the MHD reference simulation of emerging magnetic flux in a turbulent convection zone.

2nd row: The inductive components of E and Sz determined using the PTD method.

3rd row: E and Sz derived by incorporating Doppler flows around PILs into the PTD solutions. Note the dramatic improvement in the estimate of Sz.

4th row: E and Sz derived by incorporating only non-inductive FLCT derived flows into the PTD solutions. Note the poorer recovery of Ex, Ey and Sz relative to the case that included only Doppler flows.

5th row: E and Sz derived by including both Doppler flows and non-inductive FLCT flows into the PTD solutions. Note the good recovery of Ex, Ey, and Sz, and the reduction in artifacts in the low-field regions for Ey.

Qualitative and quantitative comparisons show good recovery of the simulation’s E-field and Poynting flux Sz.

Figure 3. Upper left: A comparison of the vertical component of the Poynting flux derived from the PTD method alone with the actual Poynting flux of the MHD reference simulation.

Upper right: A comparison between the simulated results and the improved technique that incorporates information about the vertical flow field around PILs into the PTD solutions.

Lower left: Comparison of the vertical Poynting flux when non-inductive FLCT-derived flows are incorporated into the PTD solutions.

Lower right: Comparison of the vertical Poynting flux when both Doppler flow information and non-inductive FLCT-derived flows are incorporated into the PTD solutions. Poynting flux units are in [105 G2 km s−1]

Conclusions

• We have reviewed how vector magnetogram sequences can, by themselves, be used to estimate electric fields E and vertical Poynting fluxes Sz.

• We then presented a new method to incorporate Doppler shifts observed along polarity inversion lines (PILs) to improve accuracy of estimates for E and Sz.

• We have also shown how information from tracking methods, e.g., LCT, can also be incorporated into estimates of E and Sz.

Future Work

• We plan to test our methods further with more realistic simulations of photospheric evolution, including sensitivity to magnetogram noise.

• We plan to apply the method to observations, to investigate relationships between estimated Poynting fluxes and flares, CMEs, and coronal heating.