Incorporating Magnetogram Data into Time-Dependent Coronal Field

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					Incorporating Magnetogram Data into
Time-Dependent Coronal Field Models
   By George Fisher, Bill Abbett, Dave Bercik,
       Jim McTiernan, and Brian Welsch
                 Space Sciences Laboratory,
               University of California, Berkeley

Abstract: We briefly review our efforts to incorporate sequences
of photospheric vector magnetograms into MHD simulations of
coronal evolution, in an effort to create data-driven models of
the coronal magnetic field. Such models should improve our
understanding of flares and coronal mass ejections (CMEs), and
might eventually lead to predictive capabilities.
How can we predict the onset of flares & CMEs?

 Free magnetic energy stored in electric currents JC in the coronal
 magnetic field is thought to drive flares & CMEs.

 Measurements of the (vector) coronal field BC , however, are rare &
 subject to large uncertainties.

 Effectively, coronal electric currents cannot be directly measured.

 Hence, current forecast methods statistically relate other data (e.g.,
 aspects of the photospheric magnetic field BP) to flares/ CMEs.
Can the essentially statistical character of current
flare & CME forecast methods be improved?

   In his review article, “Driving major solar flares
   and eruptions”, Schrijver (2009) notes:
     “Whether deterministic forecasting is in principle
     possible remains to be seen: to date no reliable
     such forecasts can be made.”


    What capabilities must be developed before
    deterministic predictions can be made?
Two possible developments might enable
deterministic forecasting.
  (i) An observational signature of imminent eruption
      could be discovered.
        But observers have searched long and hard for
        such a signature, without success.


  (ii) A time-dependent model of coronal magnetic field
    that incorporates data could be used to identify
    magnetic structures prone to flares/ CMEs.
        Here, we review our efforts to develop such a
        model.
NB: Key aspects of flares and CMEs are “known
unknowns” (Rumsfeld 2002).
• The magnetic structure of flare- & CME-prone coronal field
  configurations is unknown. For instance:

   – Are magnetic nulls essential for flares / CMEs?
   – Must twisted flux ropes exist prior to eruptions?


• What triggers the sudden release of stored free magnetic
  energy in the corona in flares & CMEs?


  Data-driven coronal models must first be used for
  interpretation before we can make predictions!
Key components of a time-dependent, data-driven
model of the coronal magnetic field BC(x,y,z; t) are:

 1. A initial state BC(x,y,z; t0) derived from data.

 2. A method to incorporate observed data to
    drive the model forward for t > t0, such as:
    – measurements of the photospheric field BP
    – possibly, coronal EUV or X-ray observations


 3. A model that can accurately simulate coronal
    evolution in response to driving.
Step 1. For the initial coronal field BC, we use a non-
linear force-free field (NLFFF), with |JC(x,y,z;t0)|≠ 0.

 • Some jargon:
    -   Force-free field: JC x BC = 0; current JC is parallel to BC
    -   Alpha: coefficient function between JC & BC, JC = αBC
    -   Linear Force-free field: α is constant in space
    -   Non-linear Force-free field: α varies in space
    -   Potential field: α = 0, JC = 0; also called “current-free”


   Since free energy emerges with active region fields
   (Leka et al. 1996), a realistic initial state will, in
   general, include currents: |JC( x, y, z; t0)|≠ 0.
We extrapolate NLFFFs via the Optimization Method
of Wheatland, Roumeliotis, and Sturrock (2000).
 The Optimization starts from an ambiguity-resolved
 vector magnetogram of the photospheric field BP.
Ideally, the magnetogram        29 Oct. 2003, 18:46 UT
would be recorded in a
force-free layer of the solar
atmosphere.

The solar photosphere is
generally not force-free, so
extrapolations from the
“forced” BP might not
represent BC accurately.

(The chromospheric
magnetogram shown here is
approximately force-free.)



                                           IVM data from T. Metcalf
 The initial field for the Optimization Method is a po-
 tential field extrapolated from the photospheric Bz.
                                                           AR 10486, 29 Oct. 2003
This image shows field lines
from the extrapolated
potential field.

The input magnetogram need
not be flux balanced, but no
current flows on flux that
leaves the box.

Note that these field lines do
not appear sheared --- they
cross polarity inversion lines
(PILs) at nearly right angles ---
cf., the next figure!



                                    The brighter field lines have stronger Bz at the surface.
 NLFFF fields resulting from the Optimization Method
 differ from the (unique) potential field.
This image shows field             AR 10486, 29 Oct. 2003
lines from the
extrapolated NLFFF.

Some field lines here do
appear sheared --- they
do cross PILs at acute
angles, and in some cases
run nearly parallel to
them.

Note the helical character
of some field lines.
Step 2. We use observations of BP(x,y,0; t) for t>t0 to derive
time-dependent boundary conditions to evolve BC(x,y,z; t).

• Faraday’s Law enables estimation the photospheric
  electric field E(x,y,0; t) from evolution of BP(x,y,0; t).

   – “Component driving” method exist to estimate E from Bz/t,
     e.g., ILCT (Welsch et al. 2004), MEF (Longcope 2004)
   – Here, we describe the “PTD” method “vector driving,” i.e.,
     inferring E from evolution of the full magnetic vector B/t


• Assuming the resistive component R of the electric field E
  can be determined from the model’s current state, Ohm’s
  Law can be used to determine v: E = -(v x B)/c + R (7)
We use a poloidal-toroidal decomposition (PTD) of
the vector BP/t to derive E.
   A divergence-free (solenoidal) vector field can be decomposed
   via
                 Ý            Ý        Ý
                B          Jz (8),
                              zˆ        ˆ
   where the overdot represents the partial time derivative. From
   (8), three scalar potentials can be derived:
                    Ý Ý
                   2       (9);
                    h       z

                    2Ý
                                Ý
                            4 Jz
                  h J               ˆ         Ý
                                    z  ( h  Bh ) (10);
                              c
                       Ý
                    2 
                           
                                    Ý
                   h    h  Bh (11).
                      z 
   When solving these Poisson equations, care must be taken with
   boundary conditions! See Fisher et al. (“nearly submitted”).
           
Faraday’s Law relates the electric field E related to
the PTD scalar potentials,
               1  1
                       Ý
                                 Ý  1  2 ˆ (12)
                                  ˆ        Ý
          E   h    h  Jz      h z
               c    z  c        c
           1
        E
           c
                          
               h  z  Ýz    EI   (13)
                    ݈ J ˆ

   To derive (13), we have “uncurled” (12), and
   consequently had to introduce the (unknown) gradient
 of a scalar potential ψ.


  Since the electric field arising from ψ is derived from a
  gradient, it has no curl, and magnetic evolution does not
  directly constrain ψ – other information must be used!
Does it work? Here’s a comparison of B/t from the
MHD data analyzed by Welsch et al. (2007).
   Bx/t ANMHD        By/t ANMHD        Bz/t ANMHD




   Bx/t derived      By/t derived      Bz/t derived




                                                            15
So the magnetic evolution is reproduced, but
how well is E recovered? First, try setting ψ=0:
       Ex true              Ey true           Ez true




      Ex derived         Ey derived       Ez derived




                                                        16
Hence, B/t enables recovery of the curl of E, but
recovering E itself requires accurately estimating ψ.

We have investigated two approaches to specify ψ:

(a) A relaxation algorithm finds ψ consistent with ideal evol-
   ution, (E-R)B = ψB; but this doesn’t fully constrain ψ.

(b) A variational method, which can also derive a ψ con-
  sistent with ideal evolution, as well as Longope’s MEF:
                     I  2             2           2 
   min      dxdyW  x 
                     2
                     (E
                         x
                              )  (E y 
                                     I

                                         y
                                            )  (E z 
                                                   I
                                                         )  (14)
                                                       z 

Further details (many!) are provided in Fisher et al. (2009)
Modeling the photosphere to corona is challenging:
  time scales are rapid; length scales are short; radiative transfer
  plays a key role; temperature & density are highly stratified.


Computational limitations preclude faithful modeling of
all of these processes on active region length scales.

Guiding question: What is the simplest algorithm that
reasonably approximates the physics of these complex
atmospheric layers over a large spatial domain?
Goal: To obtain physically meaningful results by solving
the MHD conservation equations on a discretized mesh.


      
              u   0
      t
       u                     B 2  BB      
                uu   p 
                                    I     Π   g
       t                     8  4
                                              
      B
             uB  Bu        B 
      t                                                 Currently in RADMHD:
      e                                                Red --- treated explicitly
            eu    p  u         B   Q
                                            2
                                                         Blue --- treated implicitly
      t                          4                     Purple --- combination of both



   In Abbett (2007) we developed a semi-implicit MHD model, RADMHD,
   that advances the explicit portion of the MHD system (shown above)
   by means of a third order-accurate CWENO shock capture scheme,
   and solves the implicit portion of the system via a JFNK technique.
Flows from Step 2. are incorporated into the model as a time-
dependent force --- i.e., a source term to the momentum equation.

 First, define the physical contribution to the force as defined by the MHD
 momentum conservation equation:

                               B2  BB    
                          
           Fphys      uu p   
                                       I        g (20)
                               8  4    
 Further, define the forces implied by the data:
                                 uILCT / MEF
                     Fdata                      (19)
                                  t

 Then in a single horizontal layer corresponding to the model’s photosphere,
 we recast the MHD momentum equation in the following form:
            
            u 
                   Fdata   1  Fphys   phys|| , (20)
                                                        F
             t phot
 where the parallel and perpendicular subscripts denote the forces
 parallel or perpendicular to the direction of the magnetic field.
This seemingly underwhelming simulation result is a preliminary
demonstration of feasibility of this approach.




             t=0s                                  t = 741.6 s
AR8210 IVM vector magnetogram timeseries




t=0s                              t = 741.6 s
Summary of recent improvements to RADMHD:
We have developed and implemented a computationally efficient
method of approximating optically thick radiative cooling in our
RADMHD quiet Sun models. The treatment improves upon the ad hoc
method presented in Abbett (2007), while still retaining the efficiency
necessary to allow for large, active region-scale, convection zone-to-
corona computational domains.


The simulations presented here are preliminary. Time will tell
whether the new RT treatment is robust, and whether it can maintain the
average superadiabatic stratification necessary to sustain solar-like
convective turbulence over the long timescales necessary to study the
physics of the convective dynamo. However, the initial results are
encouraging, and the simulations continue to progress.
Conclusion: Capabilities required for data-driven
modeling of the coronal magnetic field are still in
development, but progress is being made.

1. We are continuing to develop our techniques for determining
an initial state for the coronal magnetic field BC(x,y,z; t).

2. We are investigating innovative methods to specify an
electric field, E(x,y,0;t), from the evolution of the photospheric
field BP(x,y,z; t), which can be used to drive the coronal model
forward in time.

3. We have also developed a rudimentary means of assimilating
a time series of vector magnetograms into the active zones of
an MHD model in a manner that is stable, and does not over-
specify the problem.
Acknowledgements:
This ongoing work is supported by NASA, through its
Heliophysics Theory and Living With a Star TR&T
programs, and the National Science Foundation,
though its ATM, SHINE, and National Space Weather
programs. Many of the simulations presented here
were performed on NASA’s NCCS Discover
supercomputer.

Disclaimer:
Brian Welsch pressured George Fisher into presenting
this poster against George’s will; Brian is also
responsible for any inaccurate, objectionable, or
cheesy content.

				
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