A Data Driven Magnetohydrodynamic MHD Model of Active Region Evolution as a Tool for SDOHMI Data Ana

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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
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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

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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

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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.

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minimum energy method (Metcalf, 1994).
structure minimization method (Georgoulis et
al, 2004).
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
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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
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Continuity
Momentum
Viscous effect
Inertial centrifugal force
Coriolis force
Viscous dissipative function
Energy
Induction
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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
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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).
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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
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Powells 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
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Numerical Code Flow-chart
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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.
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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,tk1) and Bz(x,y,tk) are obtained
from MDI magnetograms. This process is
repeatedly carried out through the total
simulation time.
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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

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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.
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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)
Leightons 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.
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V.1 Quantification of Non-potential Field
Parameters

• Initial State
• Evolution of Lss
• Comparison between observed and simulated
  non-potential field parameters

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Evolution of Lss
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The computed 3D magnetic field evolution (left
column) and corresponding field line projection
on the x-z plane with density enhancement
contours (right column)
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V.2 Evolution of Plasma Flow and Field on the
Photosphere Surface
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Bz Contours and Transverse B-field
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Transverse Velocity and Bz Contours
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Magnetic Energy passing through the
photosphere
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Time 1251UT
Time 1427UT
Magnetic Energy (1029 erg)
Time 1603UT
Time 1739UT
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V.3 Energy Flux through the Photosphere Surface
Flow Effect
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V.4 Energy Flux through the Photosphere Surface
Flow Effect
To be determined.
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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