Using Electric Fields to Drive Simulations of the Solar Coronal Magnetic

FieldGeorge Fisher1, Mark Cheung2, Marc DeRosa2,

Maria Kazachenko1, Brian Welsch1, Todd Hoeksema3, Xudong Sun3, and the SDO vector

magnetogram team

1Space Sciences Lab, UC Berkeley2Lockheed Martin Solar and Astrophysics

Laboratory3Stanford University

SHINE Meeting ~ 25 June 2012 ~ Maui, HI

Electric Fields at the photosphere provide

critical information for driving time-dependent

models of the solar magnetic field

We illustrate this concept by using one of the simplest possible magnetic field models

possible for the solar corona, the magneto-frictional model.

Magnetofrictional Scheme

Solve

Velocity proportional to Lorentz forceimplemented via , where is a frictional coefficient that determines relaxation timescale

Was originally implemented in a solar context by Yang, Sturrock & Antiochos (1986) and Craig & Sneyd (1986)

Magnetofrictional Scheme

Our implementation evolves vector potentialvia (guarantees and also allows relative helicity to be calculated easily)

Evolved forward in time using explicit 2nd-order derivatives, spatial discretization on a Yee (1966) grid

Follows scheme pioneered by van Ballegooijen, Priest & Mackay (2000)

Magnetofrictional Scheme

Boundary conditions specified in terms of vector potential at lower boundary; sides and top boundaries are open

We take the approach of using a temporal sequence of magnetogram data to drive the simulation

Others have also used a time-dependent Boundary Conditions with a MF scheme, e.g., Yeates, Mackay & van Ballegooijen (2008)

Data-driven Modeling of AR 11158

Evolving magneto-frictional scheme enables the construction of time-dependent models of active region coronae

Modeled fields respond to photospheric driving and energy inputs

Allows measurement of buildup of free energy and helicity in response to such driving

AR 11158 on disk from 2011 February 10 – 19 or so

Lower Boundary

Need to determine at lower boundary

Here, we use that were determined by and using methods of Fisher et al. (2012), which make use of time series of Doppler and vector field measurements from HMI. See poster by Kazachenko et al. at this meeting.

AR11158 produced an X2.2 flare on 2011-02-15 01:45

Electric Field Inversion

We decompose the partial time derivative of the magnetic vector field B into two unknown functions, the Poloidal and Toroidal potentials. This is the

origin of the name Poloidal-Toroidal Decomposition, or PTD.

Each of the three variables of the right-hand equation above obeys a 2-d Poisson equation, which depends directly on the magnetic

data:

See Fisher et al. 2010 (ApJ 715, 242)

Electric Field Inversion

From which we infer that:

The potential function(s) are very important! Without the potential functions, the PTD electric field does a poor job of reconstructing the actual field in MHD simulation test cases. Currently, we include the following terms in our

inversions: The second and third terms represent non-inductive

contributions from Doppler shifts, and pattern-motions (derived from e.g. FLCT or DAVE), respectively, from

which the inductive contributions have been

removed. The fourth term is used to enforce the condition

EB=0.

See Fisher et al. 2012, Sol. Phys. 277, p153 for details. More information about these

techniques can be found in Maria Kazachenko’s poster in the 2012 SHINE meeting.

Faraday’s law says:

Caveats and cautions

Consider these as numerical experiments

No momentum or energy equation solved (no concern for heating, radiative losses, etc.)

Cartesian domain (curvature ignored)

Initialized using potential field

AR 11158

HMI line-of-sight magnetograms, remapped onto a co-rotating Cartesian reference frame

Emissivity of fieldlines is prop. to

AR 11158 modeltop view views from sides

Emissivity of fieldlines is prop. to

AR 11158 modeltop view views from sides

Emissivity of fieldlines is prop. to

AR 11158 modeltop view views from sides

AR 11158 model

Free energy of order 15% or 20% of potential field

free energy

pot’l energy

total energy

AR 11158 model

R

X2.2 flare

Summary

Magnetofriction provides a way to perform numerical experiments of data-driven coronal magnetic field evolution

Applied scheme to AR11158, using time series of HMI Doppler and vector magnetogram data to drive the model

Can study buildup of free energy and helicity in the model and the possible ejection of flux