Download Modeling the Dynamic Solar Atmosphere:

Update: Incorporating Vector Magnetograms
into Dynamic Models of the Solar Atmosphere
CISM-AG Meeting: March 2006
Bill Abbett, Brian Welsch, George Fisher
SSL, UC Berkeley
The objective:
•
To directly incorporate observations of the vector magnetic field at the
photosphere (or chromosphere) into physics-based dynamic models
of the solar atmosphere
The requirements:
(1) Sequences of reduced, ambiguity-resolved vector magnetograms
of sufficient quality to incorporate into an MHD code
(2) A robust method of determining the electric field consistent with both
the observed evolution of the photospheric field and Faraday’s Law
(3) An MHD code (or set of coupled codes) capable of modeling a region
encompassing the photosphere (where relatively reliable measurements
of the magnetic field are available), chromosphere, transition region and
corona
(4) A physically self-consistent means of incorporating (1) and (2) into (3)
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(1) Understand sequences of reduced, high-quality, ambiguity-resolved
vector magnetograms well enough to incorporate them into a numerical
simulation
•
What magnetic field data provide the most important information about
the state of the solar atmosphere, and how do we prepare the data and
make best use of it?
•
What is the best way to generate the initial atmosphere of a
time-dependent calculation; one that is both physically meaningful,
and consistent with the relevant observations of the corona?
(current method: the “optimization technique”, e.g., Wheatland et al. 2000)
•
How do we best describe the evolution of a model photosphere
given the evolution of, and noise in, the observed data and our best
understanding of the most important physics?
We currently rely on our CISM colleagues, and our SHINE-funded
collaboration with CoRA and MSU to obtain quality measurements of active
region vector magnetic fields, and to address each of these questions prior
to attempting to incorporate a given dataset into a numerical calculation.
(e.g., IVM data: AR8210, May 1998; AR9046, June 2000; AR10030, July 2002; AR10725, Feb 2005)
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(2) A method of determining the electric field consistent with both the
observed evolution of the photospheric field and the MHD induction
equation: e.g., ILCT (Welsch et al. 2004)
•
Apply Fourier Local Correlation Tracking (FLCT, Welsch et al. 2004) to
to obtain an approximation to the 2D flux transport velocity uf
•
Note that uf does not represent the 3D flow field of the magnetized
plasma, v. However, the two are geometrically related (Demoulin &
Berger 2003):
Bnu f  Bn v t  B t vn
Note that FLCT and ILCT are CISM deliverables
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To demonstrate how ILCT relates the MHD induction equation to the flux
transport velocity, consider the vertical component of the ideal MHD
induction equation (here, for clarity, we neglect the resistive term --- in
general, it can be included):
Bn
  t  Bn v t  vn B t   0
t
Substituting the geometric relation of the previous slide, we have:
Bn
  n  Bnu f   0
t
(1)
Now, simply define Bnuf in the following way:
Bnu f  t   t   n 
(2)
Substituting this expression into (1) yields
Bn
  t2
t
Since the LHS is known, we
have a Poisson equation for φ
that can be easily solved.
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Taking the curl of (2), we have
 t  Bnu f  n  t2
If we assume that u(FLCT) (our LCT approximation of uf) represents a true
flux transport velocity, we again have a solvable Poisson equation. With
both scalar potentials known, we can determine a flux transport velocity that
is both consistent with the observed evolution of the photospheric field and
the MHD induction equation:
Bnu f  t   t   n 
Up to this point, the analysis only requires the normal component of
the magnetic field! The vector field is necessary only when extracting
the 3D flow field from
B t vn
u f  vt 
Bn
Note that to obtain v, we must appeal to the fact that field-aligned flows
are unconstrained by the induction equation (one way of closing the system
is to simply assume v B  0 ).
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Brian Welsch has recently implemented a preliminary, automated
“Magnetic Evolution Pipeline” (MEP):
•
New MDI magnetograms are automatically downloaded (cron
checks for new magnetograms using wget), de-projected, and
tracked using FLCT
•
The output stream includes de-projected magnetograms, FLCT
flows (.png graphics files and ASCII data files), and tracking parameters
•
Full documentation and all codes (including possible bugs!) are
currently online
http://solarmuri.ssl.berkeley.edu/~welsch/public/data/Pipeline/
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(3) An MHD code (or set of coupled codes) capable of modeling a
region encompassing the photosphere, chromosphere, transition
region and corona
Some realities:
Extreme spatial and temporal disparities
• small-scale, active region, and global features are fundamentally interconnected
• magnetic features at the photosphere are long-lived (relative to the
convective turnover time) while features in the magnetized corona can
evolve rapidly (e.g., topological changes following reconnection events)
Vastly different physical regimes
• photosphere and below: relatively dense, turbulent (high-β) plasma with
strong magnetic fields organized in isolated structures
• corona: field-filled, low-density, magnetically dominated plasma (at least
around strong concentrations of magnetic flux!)
• flow speeds in CZ below the surface are typically below the characteristic
sound and Alfven speeds, while the chromosphere, transition region and
corona are often shock-dominated
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…different physical regimes (cont’d)
•
corona: energetics dominated by optically thin radiative cooling, anisotropic
thermal conduction, and some form of coronal heating consistent with the
empirical relationship of Pevtsov et al. 2003 (energy dissipation as measured
by soft X-rays proportional to the measured unsigned magnetic flux at the
photosphere)
•
photosphere/chromosphere: energetics dominated by optically thick radiative
transitions
Additional computational challenges:
A dynamic model atmosphere extending from at or below the photosphere
to the corona must:
•
span a ~10 order of magnitude change in gas density and a thermodynamic
transition from the 1MK corona to the optically thick, cooler layers of the low
atmosphere, visible surface, and below
•
resolve a ~100km photospheric pressure scale height (energy scale height in
the transition region can be as small as 1km!) while following large-scale
evolution
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Idealized attempts to “couple” disparate regimes:
Sub-surface anelastic

Zero-β corona
(Abbett, Mikic et al. 2004)



(Abbett et al. 2005)
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Toward more realistic AR models:
We must solve the following system:
Energy source terms (Q) include:
• Optically thin radiative cooling
• Anisotropic thermal conduction
• An option for an empirically-based coronal heating mechanism --- must
maintain a corona consistent with the empirical constraint of Pevtsov
(2003)
• LTE optically thick cooling (options: solve the grey transfer equation in the
3D Eddington approximation, or use a simple parameterization that
maintains the super-adiabatic gradient necessary to initiate and maintain
convective turbulence)
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Surmounting practical computational challenges
•
•
The MHD system is solved semi-implicitly on a block adaptive mesh.
The non-linear portion of the system is treated explicitly using the semidiscrete central method of Kurganov-Levy (2000) using a 3rd-order
CWENO polynomial reconstruction
– Provides an efficient shock capture scheme, AMR is not required to resolve
shocks
•
The implicit portion of the system, the contributions of the energy source
terms, and the resistive and viscous contributions to the induction and
momentum equations respectively, is solved via a “Jacobian-free”
Newton-Krylov technique
– Makes it possible to treat the system implicitly (thereby providing a means to
deal with temporal disparities) without prohibitive memory constraints
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Quiet Sun relaxation run (serial test):
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Toward AR scale: MPI-AMR relaxation run (test)
The near-term plan:
• Dynamically and energetically
relax a 30Mm square Cartesian
domain extending to ~2.5Mm
below the surface.
• Introduce a highly-twisted ARscale magnetic flux rope (from
the top of a sub-surface
calculation) through the bottom
boundary of the domain
• Reproduce (hopefully!) a
highly sheared, δ-spot type AR at
the surface, and follow the
evolution of the model corona as
AR flux emerges into, reconnects
and reconfigures coronal fields
Q: How do different treatments of the
coefficient of resistivity, or changes in
resolution affect the topological evolution of
the corona?
The long term plan:
• global scales / spherical
geometery
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(4) Towards a physically self-consistent means of incorporating
(1) and (2) into (3): The “Active Boundary Layer”
Use AMPS as essentially two, fully coupled codes: a thin, dynamic
photospheric layer actively coupled (internally; i.e., not via a framework
such as InterComm) to the AMPS domain
• Within the thin, photospheric boundary layer, the continuity, induction,
and energy equations are solved given an ILCT flow field (assumed to
permeate the entirety of the thin layer).
• This active boundary is dynamically coupled to AMPS, which solves
the full MHD system in a domain that extends from the top of the
model photosphere into the transition region and low corona
Inherent physical assumption: Coronal forces do not affect the photosphere
This internally-coupled system could instead extend to the low
transition region, and then be externally coupled (e.g., via InterComm) to
existing Coronal models whose lower boundaries necessarily reside in
the transition region.
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Data Driving --- The Strategy:
Model Corona
“Active” Boundary Layer
Observational Data / ILCT
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Summary (where we’re at):
(1) Sequences of reduced, ambiguity-resolved vector magnetograms of
sufficient quality to incorporate into an MHD code

--- we look forward to the increasing availability of sequences of quality vector magnetograms
(2) A robust method of determining the electric field consistent with both
the observed evolution of the photospheric magnetic field and
Faraday’s Law

--- complete
(3) An MHD code (or set of coupled codes) capable of modeling a region
encompassing the photosphere (where relatively reliable measurements
of the magnetic field are available), chromosphere, transition region and
low corona

--- almost there….
(4) A physically self-consistent means of incorporating (1) and (2) into (3)

--- still working on it! Hope to have something to present at SHINE…
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