1. Update: Incorporating Vector Magnetograms into Dynamic Models of the Solar Atmosphere CISM-AG Meeting: March 2006 Bill Abbett, Brian Welsch, George Fisher SSL, UC Berkeley

2. The objective: To directly incorporate observations of the vector magnetic field at the photosphere (or chromosphere) into physics-based dynamic models of the solar atmosphere The requirements: (1) Sequences of reduced, ambiguity-resolved vector magnetograms of sufficient quality to incorporate into an MHD code (2) A robust method of determining the electric field consistent with both the observed evolution of the photospheric field and Faraday’s Law (3) An MHD code (or set of coupled codes) capable of modeling a region encompassing the photosphere (where relatively reliable measurements of the magnetic field are available), chromosphere, transition region and corona (4) A physically self-consistent means of incorporating (1) and (2) into (3) 1

3. (1)Understand sequences of reduced, high-quality, ambiguity-resolved vector magnetograms well enough to incorporate them into a numerical simulation What magnetic field data provide the most important information about the state of the solar atmosphere, and how do we prepare the data and make best use of it? What is the best way to generate the initial atmosphere of a time-dependent calculation; one that is both physically meaningful, and consistent with the relevant observations of the corona? (current method: the “optimization technique”, e.g., Wheatland et al. 2000) How do we best describe the evolution of a model photosphere given the evolution of, and noise in, the observed data and our best understanding of the most important physics? We currently rely on our CISM colleagues, and our SHINE-funded collaboration with CoRA and MSU to obtain quality measurements of active region vector magnetic fields, and to address each of these questions prior to attempting to incorporate a given dataset into a numerical calculation. (e.g., IVM data: AR8210, May 1998; AR9046, June 2000; AR10030, July 2002; AR10725, Feb 2005) 2

4. (2) A method of determining the electric field consistent with both the observed evolution of the photospheric field and the MHD induction equation: e.g., ILCT (Welsch et al. 2004) Apply Fourier Local Correlation Tracking (FLCT, Welsch et al. 2004) to to obtain an approximation to the 2D flux transport velocity u f Note that u f does not represent the 3D flow field of the magnetized plasma, v. However, the two are geometrically related (Demoulin & Berger 2003): Note that FLCT and ILCT are CISM deliverables 3

5. To demonstrate how ILCT relates the MHD induction equation to the flux transport velocity, consider the vertical component of the ideal MHD induction equation (here, for clarity, we neglect the resistive term --- in general, it can be included): Substituting the geometric relation of the previous slide, we have: Now, simply define B n u f in the following way: Substituting this expression into (1) yields (1) Since the LHS is known, we have a Poisson equation for φ that can be easily solved. (2) 4

6. Note that to obtain v, we must appeal to the fact that field-aligned flows are unconstrained by the induction equation (one way of closing the system is to simply assume ). Taking the curl of (2), we have If we assume that u (FLCT) (our LCT approximation of u f ) represents a true flux transport velocity, we again have a solvable Poisson equation. With both scalar potentials known, we can determine a flux transport velocity that is both consistent with the observed evolution of the photospheric field and the MHD induction equation: Up to this point, the analysis only requires the normal component of the magnetic field! The vector field is necessary only when extracting the 3D flow field from 5

7. Brian Welsch has recently implemented a preliminary, automated “Magnetic Evolution Pipeline” (MEP): New MDI magnetograms are automatically downloaded (cron checks for new magnetograms using wget), de-projected, and tracked using FLCT The output stream includes de-projected magnetograms, FLCT flows (.png graphics files and ASCII data files), and tracking parameters Full documentation and all codes (including possible bugs!) are currently online http://solarmuri.ssl.berkeley.edu/~welsch/public/data/Pipeline/ 6

8. (3)An MHD code (or set of coupled codes) capable of modeling a region encompassing the photosphere, chromosphere, transition region and corona Some realities: Extreme spatial and temporal disparities small-scale, active region, and global features are fundamentally inter- connected magnetic features at the photosphere are long-lived (relative to the convective turnover time) while features in the magnetized corona can evolve rapidly (e.g., topological changes following reconnection events) Vastly different physical regimes photosphere and below: relatively dense, turbulent (high-β) plasma with strong magnetic fields organized in isolated structures corona: field-filled, low-density, magnetically dominated plasma (at least around strong concentrations of magnetic flux!) flow speeds in CZ below the surface are typically below the characteristic sound and Alfven speeds, while the chromosphere, transition region and corona are often shock-dominated 7

9. …different physical regimes (cont’d) corona: energetics dominated by optically thin radiative cooling, anisotropic thermal conduction, and some form of coronal heating consistent with the empirical relationship of Pevtsov et al. 2003 (energy dissipation as measured by soft X-rays proportional to the measured unsigned magnetic flux at the photosphere) photosphere/chromosphere: energetics dominated by optically thick radiative transitions Additional computational challenges: A dynamic model atmosphere extending from at or below the photosphere to the corona must: span a ~10 order of magnitude change in gas density and a thermodynamic transition from the 1MK corona to the optically thick, cooler layers of the low atmosphere, visible surface, and below resolve a ~100km photospheric pressure scale height (energy scale height in the transition region can be as small as 1km!) while following large-scale evolution 8

10. Idealized attempts to “couple” disparate regimes:    Sub-surface anelasticZero-β corona  (Abbett et al. 2005) (Abbett, Mikic et al. 2004) 9

11. Idealized dynamic calculations (no explicit coupling): Left: Magara (2004) ideal MHD AR flux emergence simulation as shown in Abbett et al. 2005 Right: Manchester et al. (2004) BATS-R-US MHD simulation of AR flux emergence 10

12. Toward more realistic AR models: We must solve the following system: Energy source terms (Q) include: Optically thin radiative cooling Anisotropic thermal conduction An option for an empirically-based coronal heating mechanism --- must maintain a corona consistent with the empirical constraint of Pevtsov (2003) LTE optically thick cooling (options: solve the grey transfer equation in the 3D Eddington approximation, or use a simple parameterization that maintains the super-adiabatic gradient necessary to initiate and maintain convective turbulence) 10

13. Surmounting practical computational challenges The MHD system is solved semi-implicitly on a block adaptive mesh. The non-linear portion of the system is treated explicitly using the semi- discrete central method of Kurganov-Levy (2000) using a 3 rd -order CWENO polynomial reconstruction –Provides an efficient shock capture scheme, AMR is not required to resolve shocks The implicit portion of the system, the contributions of the energy source terms, and the resistive and viscous contributions to the induction and momentum equations respectively, is solved via a “Jacobian-free” Newton-Krylov technique –Makes it possible to treat the system implicitly (thereby providing a means to deal with temporal disparities) without prohibitive memory constraints 11

14. Quiet Sun relaxation run (serial test): 12

15. Toward AR scale: MPI-AMR relaxation run (test) The near-term plan: Dynamically and energetically relax a 30Mm square Cartesian domain extending to ~2.5Mm below the surface. Introduce a highly-twisted AR- scale magnetic flux rope (from the top of a sub-surface calculation) through the bottom boundary of the domain Reproduce (hopefully!) a highly sheared, δ-spot type AR at the surface, and follow the evolution of the model corona as AR flux emerges into, reconnects and reconfigures coronal fields The long term plan: global scales / spherical geometery Q: How do different treatments of the coefficient of resistivity, or changes in resolution affect the topological evolution of the corona? 13

16. (4) Towards a physically self-consistent means of incorporating (1) and (2) into (3): The “Active Boundary Layer” Use AMPS as essentially two, fully coupled codes: a thin, dynamic photospheric layer actively coupled (internally; i.e., not via a framework such as InterComm) to the AMPS domain Within the thin, photospheric boundary layer, the continuity, induction, and energy equations are solved given an ILCT flow field (assumed to permeate the entirety of the thin layer). This active boundary is dynamically coupled to AMPS, which solves the full MHD system in a domain that extends from the top of the model photosphere into the transition region and low corona Inherent physical assumption: Coronal forces do not affect the photosphere This internally-coupled system could instead extend to the low transition region, and then be externally coupled (e.g., via InterComm) to existing Coronal models whose lower boundaries necessarily reside in the transition region. 14

17. Data Driving --- The Strategy: Observational Data / ILCT “Active” Boundary Layer Model Corona 15

18. Summary (where we’re at): (1) Sequences of reduced, ambiguity-resolved vector magnetograms of sufficient quality to incorporate into an MHD code  --- we look forward to the increasing availability of sequences of quality vector magnetograms (2) A robust method of determining the electric field consistent with both the observed evolution of the photospheric magnetic field and Faraday’s Law  --- complete (3) An MHD code (or set of coupled codes) capable of modeling a region encompassing the photosphere (where relatively reliable measurements of the magnetic field are available), chromosphere, transition region and low corona  --- almost there…. (4) A physically self-consistent means of incorporating (1) and (2) into (3)  --- still working on it! Hope to have something to present at SHINE… 16