Construction of 3D Active Region Fields and Plasma Properties using Measurements (Magnetic Fields & Others) S. T. Wu, A. H. Wang & Yang Liu 1 Center for Space Plasma & Aeronomic Research and 2 Department of Mechanical and Aerospace Engineering The University of Alabama in Huntsville, Huntsville, Alabama USA 3 W.W. Hansen Experimental Physics Laboratory Stanford University, Stanford, CA USA Presentation at SDO Science Team Meeting, March 25-28, 2008, Napa, CA

Table of Contents I.Description of the MHD Model A.Governing Equations, Boundary Conditions and Code B.Model Inputs C.Model Outputs D.Examples – AR8100 and AR8210 II.Model Tests A.Initial Potential Fields (PF) B.Initially Observed Non-Linear Force-Free Fields (NLFFF) C.Initial Analytical NLFFF (B. C. Low’s Solution) III.Open for Suggestions

I. Description of the MHD Model A.Governing Equations, Boundary Conditions and Code Governing Equations A set of standard compressible, resistive MHD equations with higher order transport. Boundary Conditions Top and side boundary condition are non-reflective (i.e. linear extrapolation). Bottom boundary conditions are evolutionary boundary conditions obtained from the method of characteristics shown on the next slide.

Expressions derived from the method of characteristics for the physical parameters of pressure, density, the components of velocity, and magnetic field vary with time on the lower boundary are:

where the coefficients A_, B_, and C_ are given below.

Alfvén speed Fast MHD wave speed Slow MHD wave speed with Sound speed

Computational Flow Chart for the 3D MHD code where “F” and “T” represent the “false” and “true”, respectively. Note, the upper box represents the code to compute the equilibrium solution and the lower box is for computing the evolutionary solution. Numerical Code Flow Chart Trial Values Iteration F T Error Check Initial State Bottom Bdy Conditions Predict Step Top & Side Bdy Conditions Bottom Bdy Conditions Correction Step Top & Side Bdy Conditions Artificial Dissipation Time =T SAVE FT Save Data Time ≥T STOP Terminating Code MHD Equilibrium Solution Evolution Solution Drivers: Magnetic field measurements Differential rotation  0 Meridinal flow

B.Model Inputs For our examples, the model inputs are observed vector magnetograms. In principle, the model inputs could be all available observed physical quantities such as magnetic field, density, temperature and velocity.

C.Model Outputs Primary outputs – 8 physical quantities, Extended outputs – Energy, helicity and non-potentialities.

D. Examples

The simulated initial state of AR8100 at 14:27 UT 1997, Oct, 31; (a) the transverse magnetic field vector (5 G  |B t |  6 G) and contours of the line- of-sight magnetic field (B z ) with the solid lines and broken lines representing the positive and negative polarity, respectively. The color bar on the upper right side indicates the strength of the line-of- sight magnetic field (-10 G  B z  10 G) contours, (b) the transverse velocity (maximum is 1.9 km s -1 ) and B z contours, (c) the density contours at surface with transverse magnetic field, and (d) the plasma beta [  = (16  nkT)/B 2 ] distribution on the surface. The color bar at the lower right side is for both density and  contours. Active Region 8100

The simulated evolution of magnetic field at 14:27 UT, 16:03 UT, 17:39 UT and 19:12 UT Oct 31, The representation is similar to Figure 1. The color bar on the right-hand side indicates the strength of LOS magnetic field. The white arrows represent the non-potential transverse magnetic field, and black arrows represent the potential transverse magnetic field. Active Region 8100

The evolution of the vertical velocity (km s -1 ) (color coded on the right-hand side) of AR The color bar on the right-hand side represents the magnitude of the vertical velocity, where the positive- polarity region (solid lines) give the upward velocity, and the negative region (dotted lines) gives downward velocity. Active Region 8100

The simulated evolution of surface transverse velocity vector (V t ), and the contours of the line-of- sight magnetic field for AR 8100 on Oct 31, 1997; (a) at 14:27 UT with kms -1   V t   1.9 km s -1 (b) at 16:03 UT with kms -1   V t   3.7 kms -1, (c) at 17:39 UT with kms-1   V t   5.0 kms -1 and (d) at 19:12 UT with kms -1   V t   7.1 kms -1. Active Region 8100

The magnetic energy (10 22 erg/km -2 ) across the low boundary to the AR8100. Active Region 8100

Simulated energy flux through the photosphere for the Active Region AR8100. Active Region 8100

Active Region 8210

Nonpotential magnetic parameters

HINODE Event December 12,13, 2006 AR Line-of-sight current helicity (G 2 /m) pre-flare Line-of-sight current helicity (G 2 /m) post-flare

HINODE Event December 12,13, 2006 AR 10930

II. Model Tests A.Initial Potential Field (PF) B.Initially Non-Linear Force-Free Field (NLFFF) C.Initial Analytical NLFFF (B.C. Low’s Solution)

MHD Magnetic Field Configuration for AR8210 PF NLFF

Active Region 8210 Initial NLFFF Initial PF

Initial NLFFF

Magnetic Energy Analysis Table Magnetic Energy Ratio AR 8210 (3D) Analytic Solution (B. C. Low) Force- Free/Potential MHD/Potential MHD/Force-free CPU  Computer 1: a Compaq/HP DS20 Alphaserver with dual 667 MHz Alpha processors, 1 GB of RAM, and 200 GB of disk storage. 2  VA : For dimension 121 x 121 x 21 ~ 7 CPU For dimension 121 x 121 x 121 ~ 45 CPU  Computer 2: a 12-node PSSC P4 PowerWulf Beowulf cluster, based on P4 2.8 GHz processors with 40 GB storage per node and 240 GB head node storage. 2  VA : For dimension 121 x 121 x 21 ~ 4 CPU For dimension 121 x 121 x 121 ~ 25 CPU For dimension 320 x 320 x 64 ~ 56 CPU

III. Open for Suggestions

Thank You