S. T. Wu 1,2 , A. H. Wang 1 , Yang Liu 3 , and Todd Hoeksema 3

1. A Data Driven Magnetohydrodynamic (MHD) Model of Active Region Evolution as a Tool for SDO/HMI Data Analyses S. T. Wu1,2, A. H. Wang1, Yang Liu3, and Todd Hoeksema3 1Center for Space Plasma & Aeronomic Research and 2Department of Mechanical & Aerospace Engineering The University of Alabama in Huntsville, Huntsville, Alabama 35899 USA 3W.W. Hansen Experimental Physics Laboratory, Stanford University, Stanford, CA 94305-4085 HMI/AIA Science Team Meeting, Monterey, CA , February 2006

2. Table of Contents I. Motivation II. Objective • Model IV. Procedures of the Numerical Simulation and Products • Initializing the simulation of an Active Region (AR) • Evolutionary simulation of an Active Region (AR) • Example:(1997 Oct 31 – Nov 4) AR1800 • Remarks

3. I. Motivation To understand the sources of solar eruptive phenomena requires a knowledge of the evolution of the active region. Recently, there are many 3D, time-dependent magnetohydrodynamic (MHD) models and all of them are focused on global scale dynamics. But, the photospheric surface driven mechanisms such as differential rotation and meridional flow have been ignored. In order to fill this gap, a 3D, time-dependent, MHD model to describe the photospheric magnetic field transport with MHD effect is proposed. Further, the projected characteristics are fully implemented at the lower boundary, this enabling us to input photospheric observations such that the sub-photospheric (i.e. convective zone) effects can be taken into account. This model is aiming to perform data analyses for the MDI,TRACE data and upcoming SDO/HMI data

4. II. Objective • To utilize our 3D MHD model for the Active Region (AR) evolution by inputs of observables such as photospheric level measured magnetic field, velocity field and density, if these measurements are available. • In the meantime, the available measurements are limited to line-of-sight magnetic field at the photosphere. Therefore, we will input the measured LOS field together with the source surface potential field model and density model to drive the model to obtain the following physical properties. • The velocity field at photospheric level. • The 3D coronal magnetic field evolution driven by the inputs of the photopheric magnetic field. • The physics we intend to understand are: • The initiation of solar eruptions • Helicity flux through the photospere to the corona • The photospheric surface flow effects on the energy transport from the photosphere to the corona • The growth and decay of an AR.

5. “structure minimization” method (Georgoulis et al, 2004). “minimum” energy method (Metcalf, 1994). 1. Optimization approach (Wheatland et al, 2000); 2. Boundary element method (Yan & Sakurai, 2000). 1. ILCT (Welsch et al, 2004), 2. “The minimum energy fit” (Longcope, 2004), 3. “induction equation” (Kusano et al, 2002). Coronal magnetic fields. (NLFFF, MHD) Plasma flow. Synoptic maps. PFSS & HCCSSS models Connectivity & separatrix Heliospheric magnetic field. Energy flux through the photosphere. Helicity flux through the photosphere. MHD simulation Topology structure Dynamical evolution. Emerging flux effect Surface flow effect Solar transients Prediction of IMF & solar wind speed. Empirical estimate; MHD simulation. MHD simulation

6. III. Mathematical Description and Boundary Conditions for a Three-Dimensional, Time-dependent MHD Model III.1. Mathematical Model Solar eruptive events are manifested by a variety of signatures, from flares to CMEs. It is known that the energy that drives these events is stored in the non-potential magnetic fields of the active regions and the corona. Theoretically, the MHD processes are capable of describing the nonlinear interactions between the plasma flow and magnetic fields, which is essential for the understanding of the physical processes of Active Region evolution. Thus, a 3D, time dependent MHD model with differential rotation, meridional flow and effective diffusion as well as cyclonic turbulence to study the AR evolution is presented in the following:

7. Continuity: Momentum: Viscous effect Inertial centrifugal force Coriolis force Viscous dissipative function: Energy: Induction:

8. where, the angular velocity of solar differential rotation : the plasma density the plasma flow velocity gravitational force p: the plasma thermal pressure  Q: heat conduction the magnetic induction vector t: turbulent viscosity electric current : coefficient of the cyclonic turbulence energy source function : magnetic diffusivity : specific heat ratio

9. III.2. Boundary Conditions To simulate the active region evolutions, we have cast the set of governing equations described in Section III.1 in a rectangular coordinate system. The computational domain includes six planes (i.e. four side, top and bottom planes). The boundary conditions used for the four sides are linear extrapolation, and the top boundary is non-reflective. In order to accommodate the observations at the bottom boundary, the evolutionary boundary conditions are used. Thus, the method of projected characateristics are used for the derivation of such boundary conditions (Wu et al. 2005).

10. Numerical Methods The numerical method we used is simple TVD Lax-Friedrichs formulation. This scheme achieves the second order accuracy both temporally and spatially. To achieve second order temporal accuracy, the Hancock predictor step and corrector step are used. Predictor Step: Corrector Step:

11. Powell’s Corrective Terms Powell discovered that including B corrective terms and the corresponding characteristic divergence wave, can stabilize the solution for the TVD type algorithms. In our equations, the source terms include the following corrective terms:

12. Pre-Initial State Iteration T F Error Check Initial State Bottom Bdy Conditions Predict Step Top & Side Bdy Conditions Bottom Bdy Conditions Correction Step Top & Side Bdy Conditions Artificial Dissipation F T Save Data Time =TSAVE Terminating Code Time ≥TSTOP Numerical Code Flow-chart

13. IV. Procedures of the Numerical Simulation IV.1. Initializing the Simulation of the Active Region (AR) • Use the magnetic field data from photospheric magnetogram together with potential field model to construct a three-dimensional field configuration • Since there is no density measurement on the photosphere, we simply assume that the density distribution at the photospheric level is directly proportional to the absolute value of the magnitude of the transverse field and decreases exponentially with the scale height, thus where o and Bo are the constant reference with values Hg as the scale height, and (c) Input the results of (a) and (b) into the MHD model described in Section III to allow its relaxing to a quasi-equilibrium state. This will be our initial state for the study of the evolution.

14. IV.2. Evolutionary Simulation of an Active Region (AR) To study the evolution of an active region, the model is driven by differential rotation, meridianol flow and the measured magnetic field at the photospheric level. To input the measured magnetic field a procedure is developed. In this procedure, we take two consecutive sets of MDI magnetograms subtract one from the other, then incrementally increase at six second intervals to cover these two sets of MDI data. Specifically, the expression used are: Since we have chosen our time step to be six seconds and the MDI measurements are every 96 minutes, thus it takes 960 steps to fill the period, hence Where Bz(x,y,tk+1) and Bz(x,y,tk) are obtained from MDI magnetograms. This process is repeatedly carried out through the total simulation time.

15. Products • Quantification of the non-potential parameters which could lead to solar eruption • Photospheric Flows • Surface Flow Effects: The Energy Flux through the Photosphere • Emerging Flux Effects: Helicity Flux through the Photosphere

16. V Example: AR8100, 1997 Oct 31 – Nov 4 The SOHO/MDI field measurements of the active region have a resolution of ~ 2 arc sec with 198 × 198 pixels and a cadence of ~ 96 min. In order to assure the computational grids are compatible with the measurements, the computation domain are 99 in longitudinal direction (x), 99 in latitudinal direction (y) and 99 in height (z), respectively. To match the data with the grids, we have taken a four point average of the pixels inside the domain. On the boundary we have taken a two point average from the measurements. At the four corners, the measurements are used.

17. Before we can carry out the simulation study, we need to know two important coefficients; effective diffusivity () and cyclonic turbulence (). There are no precise theory and observations and laboratory experiments to determine these coefficients. However, there are some previous works which have discussed the choice of these two coefficients. For example,  = 160 – 300 km2 s-1 given by Parker, (1979); Leighton’s value of  is 800 – 1600 km2 s-1(1964); DeVore, et al (1985) selected  = 300 km2 s-1 for their study. Wang (1988) derived a value of  being 100 – 150 km2 s-1 on the basis of observation of sunspot decay. We notice that there is a wide range of values for the effective diffusivity. The value of cyclonic turbulence is chosen according to the scale law ( ~ /L), given by Parker, (1979) where L is the characteristic length of the sunspot, it is chosen to be 6,000 km for this study and  is 200 km2 s-1.

18. V.1 Quantification of Non-potential Field Parameters • Initial State • Evolution of Lss • Comparison between observed and simulated non-potential field parameters

21. Evolution of Lss

22. The computed 3D magnetic field evolution (left column) and corresponding field line projection on the x-z plane with density enhancement contours (right column)

24. V.2 Evolution of Plasma Flow and Field on the Photosphere Surface

25. Bz – Contours and Transverse B-field

26. Transverse Velocity and Bz Contours

27. Magnetic Energy passing through the photosphere

28. Time: 12:51UT Time: 14:27UT Magnetic Energy (1029 erg) Time: 16:03UT Time: 17:39UT

30. V.3 Energy Flux through the Photosphere: Surface Flow Effect

32. V.4 Energy Flux through the Photosphere: Surface Flow Effect To be determined.

33. VII Remarks • 1. Nature of the Code • Fortran • Additional Support Software • IDL • Computational Requirements • Alpha machine: 99  99  99 grid run 20,000 sec requires 32 hrs CPU • Requirements for the Input Data & Format of the Out Products • Input: • Output: