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 and2Department of Mechanical & Aerospace Engineering

The University of Alabama in Huntsville, Huntsville, Alabama 35899 USA3W.W. Hansen Experimental Physics Laboratory, Stanford University,

Stanford, CA 94305-4085

HMI/AIA Science Team Meeting, Monterey, CA , February 2006

Table of Contents

I. MotivationII. ObjectiveIII. ModelIV. Procedures of the Numerical Simulation and

Products– Initializing the simulation of an Active Region

(AR)– Evolutionary simulation of an Active Region

(AR)

V. Example:(1997 Oct 31 – Nov 4) AR1800VI. Remarks

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

II. Objective

1. 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.

2. 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.(a) The velocity field at photospheric level.(b) The 3D coronal magnetic field evolution driven by the inputs of the

photopheric magnetic field.3. The physics we intend to understand are:

(a) The initiation of solar eruptions(b) Helicity flux through the photospere to the corona(c) The photospheric surface flow effects on the energy transport from the

photosphere to the corona(d) The growth and decay of an AR.

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tn

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tt

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vB

Coronal magnetic fields.

(NLFFF, MHD)Plasma flow.Synoptic maps.

Solar transients

Connectivity & separatrix

Energy flux through the photosphere.

Heliospheric magnetic field.

Dynamical evolution.

Empirical estimate; MHD simulation.

Prediction of IMF & solar wind speed.

“minimum” energy method (Metcalf, 1994).

“structure minimization” method (Georgoulis et al, 2004).

1. ILCT (Welsch et al, 2004), 2. “The minimum energy fit” (Longcope, 2004), 3. “induction equation” (Kusano et al, 2002).

1. Optimization approach (Wheatland et al, 2000); 2. Boundary element method (Yan & Sakurai, 2000).

PFSS & HCCSSS models

MHD simulation Topology

structure

MHD simulation

Helicity flux through the photosphere.

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dHn

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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:

III. Mathematical Description and Boundary Conditions for a Three-Dimensional, Time-dependent MHD Model

III.1. Mathematical Model

Continuity: 0

ut

Momentum:

Energy:

22

211 uJQuppu

t

p t

uFBBpuut

uog

2

4

1

Inertial centrifugal force

roo

Coriolis force Viscous effect

Induction: SBBBut

B

2

Viscous dissipative function:

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2 2

where,

: the plasma density

:u

the plasma flow velocity

p: the plasma thermal pressure

:B

the magnetic induction vector

:J

electric current

energy source function

: the angular velocity of solar differential rotation

gravitational force

Q: heat conduction

: coefficient of the cyclonic turbulence

t: turbulent viscosity

: magnetic diffusivity: specific heat ratio

:gF

:S

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).

III.2. Boundary Conditions

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:

1

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,,21

,,,,,,

)()(21

x

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ni

nkjin

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x

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nj

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)(2

)()(,,

3

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nk

nkji

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t

x

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Corrector Step:

)()()()( 22

21

21

21

21

21

21 ,,22

121

21

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nkji

tHk

Hk

Gj

Gj

Fi

Fi

Tkji

nkji USUU

)]()()([21

21

21

21

21

21 2

121

21

,,,,LRk

LRk

LRj

LRj

LRi

LRi

nkji

Tkji HHGGFFtUU

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:

))((

0

BBu

Bu

Bu

Bu

BB

BB

BB

S

z

y

x

z

y

x

Pre-Initial State

Iteration

F TError Check

Initial State

Bottom Bdy Conditions

Predict Step

Top & Side Bdy Conditions

Bottom Bdy Conditions

Correction Step

Top & Side Bdy Conditions

Artificial Dissipation

Time =TSAVE

F T Save Data

Time ≥TSTOPTerminating Code

Numerical Code Flow-chart

IV. Procedures of the Numerical Simulation

IV.1. Initializing the Simulation of the Active Region (AR)

(a) Use the magnetic field data from photospheric magnetogram together with potential field model to construct a three-dimensional field configuration

(b) 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

gH

z

o

yxo e

B

BBzyx

2

22

0,,,

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.

,,,),,( k n

nzzoz BoyxBtyxB

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

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:

960...,,2,1,

960

,,),,( 1

ntyxBtyxB

Bk

zok

zonz

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.

IV.2. Evolutionary Simulation of an Active Region (AR)

Products

• Quantification of the non-potential parameters which could lead to solar eruption

• Photospheric Flows• Surface Flow Effects: The Energy Flux through the Photosphere

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21

• Emerging Flux Effects: Helicity Flux through the Photosphere

dSBvAdSvBAdt

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photo2

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.

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.

V.1 Quantification of Non-potential Field Parameters

• Initial State

• Evolution of Lss

• Comparison between observed and simulated non-potential field parameters

Evolution of Lss

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

o

o

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

Bz – Contours and Transverse B-field

Transverse Velocity and Bz Contours

Magnetic Energy passing through the photosphere 2B

Time: 12:51UT Time: 14:27UT

Time: 16:03UT Time: 17:39UT

Magnetic Energy (1029 erg)

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

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V.4 Energy Flux through the Photosphere: Surface Flow Effect

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2

To be determined.

1. Nature of the Code Fortran

2. Additional Support SoftwareIDL

3. Computational RequirementsAlpha machine: 99 99 99 grid run 20,000 sec requires 32 hrs CPU

4. Requirements for the Input Data & Format of the Out ProductsInput: Output:

VelocityDensityfieldBSVIDL ,,

volumeain ,,, TVBFormattedFortran

VII Remarks