1. Modeling Active Region Magnetic Fields on the Sun W.P. Abbett Space Sciences Laboratory University of California, Berkeley

2. Motivation for Detailed Studies of Active Region Evolution: The ability to predict “space weather” is becoming increasingly important as our society becomes more reliant on ground and space-based technologies that are susceptible to a flux of energetic particles, or fluctuating ionospheric currents. Examples of activities disrupted by solar and geomagnetic events: satellite operations, navigation (GPS), manned space flight, polar flights, power distribution networks, HF radio and long-line telephone communication, and pipeline operations.

3. Motivation for Detailed Studies of Active Region Evolution: CMEs are magnetically driven and many are believed to originate in the low solar corona in and around active region magnetic fields. Coronal Mass Ejections (CMEs) are among the primary drivers of space weather.

4. LASCO C3 Coronagraph: CME of Jan. 15, 1996

5. And now, a bit closer to the Sun… In the low corona (where the CME is initiated), magnetic flux is “frozen” into the plasma, and the field is essentially “line-tied” to the photosphere; evolves in response to changes to the Sun’s photospheric as such, it evolves in response to changes to the Sun’s photospheric magnetic field magnetic field. TRACE 171 Movie of the X-class flare producing NOAA active region 9906 on April 21, 2002. TRACE 171 Movie of the corona over a rotating sunspot in NOAA active region 9114 on August 9, 2000. Thus, understanding the evolution of the vector magnetic field at the photosphere is a crucial component of the effort to forecast and interpret space weather events.

6. Active Region Magnetic Fields at the Photosphere Active regions appear as bipoles, which implies they are the tops of large Omega-shaped loops which have risen through the solar convection zone and emerged into the photosphere. On average, bipoles are oriented nearly parallel to the E-W direction (Hale’s law 1919) indicating that the underlying field geometry is nearly toroidal. Hale’s law persists for years through a given solar cycle, thus the toroidal layer must lie deep in the interior in a region relatively free from convective turbulence. Full disk MDI magnetogram courtesy of Y. Liu Cauzzi et. al. (1996)

7. A Simple Picture of Active Region Magnetic Flux A toroidal flux layer persists near the “tachocline” --- where the turbulent convection zone transitions into the stable radiative zone. Portions of the toroidal layer succumb to a dynamic or magnetic instability causing a magnetically buoyant flux rope to rise through the convection zone. The top of the loop emerges from the high-beta plasma of the convection zone, through the many pressure scale heights of the photosphere, chromosphere and transition region, into the low-beta, tenuous, energetic corona.

8. Testing this Simple Picture Using Numerical Models Only recently has it become possible to routinely run 3D numerical simulations of the sub-surface evolution of active region-scale magnetic fields. In particular, a numerical solution of the system of MHD equations in the anelastic approximation allows for an exploration of the physics of active region magnetic fields in the solar interior. We can address questions such as: What is the role of fieldline twist, solar rotation, and field strength on the dynamics of flux emergence? Can simple models accurately represent the physics of sub-surface magnetic structures? From Abbett et. al. (2000) MDI Magnetogram from Cauzzi et al (1996)

9. Modeling Magnetic Fields in the Solar Convection Zone Is an interface dynamo necessary to generate active region-strength fields, and can magnetic flux be transported back to the base of the convection zone in the absence of an interface layer? Can magnetic flux tubes remain cohesive during their ascent through the turbulent convection zone? Can we determine how strong the magnetic field must be at the base of the convection zone to be consistent with the ~1000G fields observed at the photosphere? From Abbett et al. 2003

10. Coupling Photospheric Fields to a Model Corona: PFSS Given a representation of the magnetic field at or below the photosphere, can we characterize the magnetic topology of the corona? One method is to assume that the vector magnetic field can be derived from a scalar potential --- this assumption coupled with the requirement that the field be radial far from the photosphere (at the “source surface”), potential field source surface is the basis of the potential field source surface (PFSS) extrapolation. PFSS models of Li & Luhmann

11. Coupling Photospheric Fields to a Model Corona: FFF force-free Another method is to assume that the magnetic field is in a force-free equilibrium (i.e. j X B = 0). This type of extrapolation may provide an improved representation over the potential field model, particularly near dynamic active regions. FFF calculations from Y. Liu (2003)

12. Coupling Photospheric Fields to a Model Corona: MHD Another method --- the most computationally expensive technique so far --- is to numerically solve the system of ideal MHD equations. Why go to all that trouble? MHD models are able to describe the continuous continuous topological evolution of the corona, thus providing a means to e.g. follow the dynamic evolution of the open field, and to investigate proposed CME initiation mechanisms. To drive an MHD model corona, on must first overcome a set of unique challenges: 1. Unlike PFSS and FFF extrapolations, MHD models require information about the flow field at the photospheric boundary --- and that information is generally not available. 2.To resolve individual active regions while simultaneously following the global evolution of the corona requires the implementation of an adaptive, non-uniform mesh.

13. A First Attempt: From Abbett & Fisher (2003)

14. How Force-free are Emerging Structures? The emergence of modestly twisted magnetic flux tubes into the model corona. The color table is a measure of how “force-free” the atmosphere is --- the bluer the color the more the current density is aligned with the magnetic field lines. The difference in “tilt angle” between overlying field lines and those close to the lower boundary reflects the distribution of twist present in the flux rope below the surface From Abbett & Fisher (2003)

15. Another approach: Magara & Longcope (2003) and Fan (2001) simulate the emergence of highly just below twisted flux tubes positioned just below the photosphere --- information resulting from the buoyant rise of the tube through the entirety of the convective envelope is lost; however, there is no need to implement an interface boundary.

16. A Few Caveats: Note that treating the visible surface as a simple 2D layer or thin 3D slab clearly simplifies the physics of these layers. Since the goal is not to directly simulate coronal emission, and since we must make the problem computationally tractable at active region spatial scales, we do not treat (in detail) the physics of radiative transport and thermal conduction along field lines. An example of a more “realistic” calculation (Bercik, Stein, Nordlund 2002):

17. Using Observations to Drive a MHD Model Corona AR8210 is a very complex, CME producing active region. Though complex, AR8210 is a MURI candidate event --- mainly because the vector data at the photosphere is of high quality. Can we obtain a flow field that is consistent with the observed evolution of the Magnetic field? A high-cadence sequence of MDI magnetograms of AR8210 on May 1,1998 (S. Regnier, R. Canfield)

18. New Velocity Inversion Techniques: I+LCT I+LCT (Welsch & Fisher) --- Uses a combination of Local Correlation Tracking (LCT) on magnetic elements along with the MHD induction equation to obtain a self-consistent flow field. MEF (Longcope & Klapper) --- constrains the system by minimizing the spatially integrated square of the velocity field. A flow field that satisfies the vertical component of the induction equation is not necessarily unique!

19. Testing Velocity Inversion Techniques Examples of using ANMHD to test velocity inversion techniques.

20. Putting it all Together (A Work in Progress): I+LCT and MEF provide means to generate velocity fields from a sequence of vector magnetograms that can be used to update the lower boundary layers of an MHD model corona. Additionally, MHD models require an initial state, one that 1.Matches the transverse components of the magnetic field observed at the photosphere, and 2.Accurately represents the magnetic topology of the initial atmosphere Options: PF and FFF extrapolations.

21. The Global Topology of AR8210 Local calculations can go only so far! If CME initiation mechanisms depend on the magnetic topology of the global corona, local calculations performed in isolation may not tell the whole story Modeling the evolution of open field in the global corona is a critical component in the effort to understand and predict space weather --- in fact, recent studies have shown that Active regions are a significant source of the open field in times of heightened solar activity. Yohkoh SXT reverse image of a trans-equatorial loop emanating from AR8210.

22. Resolving Active Regions Embedded in a Global Corona Using the domain decomposition and adaptive mesh refinement tool PARAMESH (MacNeice et. al. 2000) to resolve active region magnetic fields in a global environment:

23. Toward Tetrahedral MHD/AMR:

24. Conclusions: “Sun to Mud” Models There is community-wide interest in the development of end-to-end coupled numerical models to be used as operational, real-time space weather forecasting tools. The least understood component of the Sun-earth system is the solar atmosphere --- where solar storms originate. To be successful, and to progress beyond idealized calculations, each of the above multi-agency, require coordinated efforts ultimately will require an observationally-based solar model that can routinely provide real time, physical boundary conditions for their models.