The%20Dynamic,%20Magnetic,%20and%20Energetic%20Connection%20Between%20the%20Convection%20Zone%20and%20Corona

1
The Dynamic, Magnetic, and Energetic Connection
Between the Convection Zone and Corona

• W.P. Abbett
• Space Sciences Laboratory
• Univ. of California, Berkeley

Flux Emergence Workshop Univ. of St. Andrews,
June 2007
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Motivation

• Understanding the physics of the solar magnetic
  field ---
• from its generation in the turbulent,
  differentially-rotating interior
• to the ultimate release of magnetic energy in
  the atmosphere in the
• form of radiation, solar wind acceleration,
  flares or CMEs ---
• requires
• an understanding of the dynamic, energetic
  and magnetic connection
• between the convective interior and corona.

Here, we present a first step toward this goal
Results from a recently developed code capable of
quantitatively describing the physical connection
between the Quiet Sun upper convection zone and
low-corona.
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• Characteristics of the convective interior
• The convection zone is high-density, high-ß,
  optically-thick, turbulent plasma.
• Magnetic fields entrained within convective
  flows at and below the visible
• surface evolve slowly compared to the coronal
  field.

From Abbett et al. 2004
From Bercik et al. 2005
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• Characteristics of the solar corona
• The corona is a low-density, low-ß,
  optically-thin, hot plasma
• Plasma entrained within coronal loops evolves
  rapidly
• compared to sub-surface structures
• The magnetically-dominated corona can store
  energy over long
• periods of time, but will often undergo
  sudden, rapid, and dramatic
• topological changes as magnetic energy is
  released.

Movies courtesy of LMSAL, TRACE LASCO consortia
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Attempts to connect these spatially and
temporally disparate, dynamic regimes Coupled
models of emerging active regions
From Abbett Fisher 2003
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Idealized dynamic calculations (no explicit
coupling)
Left Magara (2004) ideal MHD AR flux emergence
simulation as shown in Abbett et al.
2005 Right Manchester et al. (2004) BATS-R-US
MHD simulation of AR flux emergence
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Modeling the combined convection
zone-to-corona system in a physically
self-consistent way
The resistive fully-compressible MHD system of
equations
Closure relation a non-ideal equation of state
obtained through an inversion of the OPAL tables
(Rogers 2000),
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Modeling the combined convection
zone-to-corona system in a physically
self-consistent way
The source term in the energy equation,

• must include the important physics believed to
  govern the evolution of the combined system. In
  the corona, this includes
• radiative cooling (in the optically thin
  limit),
• the divergence of the electron heat flux,
• a coronal heating mechanism (if necessary).
• In the lower atmosphere at and above the visible
  surface,
• radiative cooling (optically thick)
• Below the surface in the deeper layers of the
  convective interior
• radiative cooling (in the optically thick
  diffusion limit)

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Modeling the combined convection zone-to-corona
system
We represent the source term as follows
In order to extend the spatial domain to active
region scales, we choose not to solve the
optically-thick LTE transfer equation to obtain
an expression for surface cooling, .
Instead, we choose to approximate this cooling in
a way that successfully reproduces the average
stratification and solar-like convective
turbulence of the more realistic simulations of
Bercik (2002) and Stein et al. (2003)
where
and and represent dimension-less
envelope functions that restrict each term to the
appropriate range of densities or depths in such
a way as to avoid sharp cutoffs. is
obtained from the CHIANTI atomic database.
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Modeling the combined convection zone-to-corona
system
The structure of the transition region and corona
depend strongly on the remaining non-radiative
terms in the
divergence of the electron heat flux,
and an additional coronal heating rate (if
necessary). We employ an empirically-based
description of coronal heating consistent with
the observed relationship between unsigned
magnetic flux and the power dissipated in the
atmosphere by a coronal heating mechanism,
.
The RHS of this equation represents the Pevtsov
et al. (2003) power law relationship between
X-ray luminosity and unsigned magnetic flux at
the photosphere . If we choose a
simple heating function of the form
(consistent with Lundquist et al. 2007),
we arrive at an empirically-based form of coronal
heating consistent with Pevtsovs Law
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Modeling the combined convection zone-to-corona
system
The calibrated radiative source term in
, coupled with a
constant radiative flux lower boundary condition
(on average) maintains the super-adiabatic
stratification necessary to initiate and sustain
convection. The thermodynamic structure of
the model is controlled by the energy source
terms, the gravitational acceleration and the
applied thermodynamic boundary conditions. No
stratification is imposed a priori.
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Numerical techniques and challenges

• A dynamic numerical model extending from below
  the photosphere out into the corona must
• span a 10 - 15 order of magnitude change in
  gas density and a thermodynamic transition from
  the 1 MK corona to the optically thick, cooler
  layers of the low atmosphere, visible surface,
  and below
• resolve a 100 km photospheric pressure scale
  height while simultaneously following large-scale
  evolution (we use the Mikic et al. 2005 technique
  to mitigate the need to resolve the 1 km
  transition region scale height characteristic of
  a Spitzer-type conductivity)
• remain highly accurate in the turbulent
  sub-surface layers, while still employing an
  effective shock capture scheme to follow and
  resolve shock fronts in the upper atmosphere
• address the extreme temporal disparity of the
  combined system

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RADMHD Numerical techniques and challenges

• For the Quiet Sun we use a semi-implicit,
  operator-split method.

• Explicit sub-step We use a 3D extension of
  the semi-discrete method of Kurganov Levy
  (2000) with the third order-accurate central
  weighted essentially non-oscillatory (CWENO)
  polynomial reconstruction of Levy et al. (2000).
• CWENO interpolation provides an efficient,
  accurate, simple shock capture scheme that allows
  us to resolve shocks in the transition region and
  corona without refining the mesh. The
  solenoidal constraint on B is enforced implicitly.

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RADMHD Numerical techniques and challenges

• For the Quiet Sun we use a semi-implicit,
  operator-split method

• Implicit sub-step We use a Jacobian-free
  Newton-Krylov (JFNK) solver (see Knoll Keyes
  2003). The Krylov sub-step employs the
  generalized minimum residual (GMRES) technique.
• JFNK provides a memory-efficient means of
  implicitly solving a non-linear system, and frees
  us from the restrictive CFL stability conditions
  imposed by e.g., the electron thermal
  conductivity and radiative cooling.

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RADMHD Numerical techniques and challenges

• The MHD system is solved on an adaptive,
  domain-decomposed mesh.
• Note With our numerical techniques,
  AMR is not needed to
• simulate the Quiet Sun. However,
  RADMHD has the capability
• of interfacing with the PARAMESH
  libraries (MacNeice et al. 2000)
• to provide an adaptive framework.
• Spatial disparities of the combined convection
  zone-to-corona system are addressed via the CWENO
  explicit scheme, the domain decomposition
  strategy, and AMR capability if necessary.
• Temporal disparities of the combined
  convection zone-to-corona system are addressed
  via the JFNK implicit scheme. Pre-conditioning
  is an essential requirement if one wishes to
  rapidly relax atmospheres by significantly
  exceeding the CFL limit.
• Boundary conditions of the Quiet Sun
  simulations Periodic in the transverse
  directions, constant radiative flux in through a
  closed lower boundary, open coronal boundary

On to the Results (Finally!)
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The Quiet Sun magnetic field in the model
chromosphere
Magnetic field generated through the action of a
convective surface dynamo. Fieldlines drawn (in
both directions) from points located 700 km above
the visible surface. Grayscale image represents
the vertical component of the velocity field at
the model photosphere. The low-chromosphere acts
as a dynamic, high-ß plasma except along thin
rope-like structures threading the atmosphere,
connecting strong photospheric structures to the
transition region-corona interface. Plasma-ß 1
at the photosphere only in localized regions of
concentrated field (near strong high-vorticity
downdrafts
From Abbett (2007)
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Flux submergence in the Quiet Sun and the
connectivity between an initially vertical
coronal field and the turbulent convection zone
From Abbett (2007)
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Convective reversal

• A brightness reversal with height in the
  atmosphere is a common feature of Ca II H and K
  observations of the Quiet Sun chromosphere.
• In the simulations, a temperature (or
  convective) reversal in the model chromosphere
  occurs as a result of the p div u work of
  converging and diverging flows in the
  lower-density layers above the photosphere where
  radiative cooling is less dominant.

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Gas temperature and Bz at 700 km above and
below the model photosphere
From Abbett (2007)
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Flux cancellation and the effects of resolution
The Quiet Sun magnetic flux threading the model
photosphere over a 15 minute interval. Grid
resolution 117 X 117 km Average unsigned flux
per pixel 34.5 G
Simulated noise-free magnetograms reduced to MDI
resolution (high-resolution mode) by convolving
the dataset with a 2D Gaussian with a FWHM of
0.62 or 459 km. Average unsigned flux per pixel
is now 19.9 G
Simulated noise-free magnetograms reduced to Kitt
Peak resolution. FWHM of the Gaussian Kernel is
1.0 or 740 km. Average unsigned flux per pixel
15.0 G Observed unsigned flux per pixel at Kitt
Peak 5.5 G
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How force-free is the Quiet Sun atmosphere?
Photosphere
Chromosphere
Temperature
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log B
log ß
Bz
log B
log J
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Characteristics of the Quiet Sun model atmosphere
Note Above movie is not a timeseries!
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Summary and conclusions to date

• It is possible to efficiently simulate the
  upper convection zone, photosphere, chromosphere,
  transition region and low-corona within a single
  computational domain
• If we use an empirically based coronal heating
  mechanism consistent with the Pevtsov et al.
  (2003) relationship between X-ray emission and
  magnetic flux observed at the surface, magnetic
  fields generated from a convective dynamo are
  sufficient to heat the corona to 1 MK.
• The model chromosphere exhibits a convective
  reversal --- plasma above cell centers tends to
  be cooler than the average temperature at that
  height, while plasma above the photospheric
  intergranular lanes is generally hotter. In the
  models, this is simply due to the
  work of converging and diverging flows in the
  relatively low density layers above the
  photosphere where the radiative source terms are
  less dominant.

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Summary and conclusions to date

• Analysis of magnetograph data at relatively
  low, one arcsecond resolution can lead to a
  significant underestimate of the total amount of
  unsigned magnetic flux threading the Quiet Sun
  photosphere.
• A persistent current layer is formed as the
  atmosphere transitions from a dynamic, high-ß
  regime (on average) to the more
  magnetically-dominated transition region and
  corona.
• The Quiet Sun model corona is dynamic, and is
  not everywhere force-free or even low-ß. There
  are regions of both relatively strong and
  exceptionally weak field high in the model
  corona.
• Synthetic vector magnetograms generated at the
  model photosphere and chromosphere are
  substantively different, and will yield
  substantively different results if used as a
  basis for force-free or potential field
  extrapolations. Neither extrapolation technique
  will reproduce the twisted horizontal structures
  present just above the model photosphere (and
  their associated magnetic connectivity) in the
  dynamic, often non-force-free layers of the low
  chromosphere.

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Summary and conclusions to date

• On average, the bulk of the unsigned magnetic
  flux resides below the visible surface. Of the
  flux that threads the surface, most remains
  entrained in the plasma, and turns over within a
  local chromospheric pressure scale height. Thus,
  there is a steep decline in the average amount of
  unsigned magnetic flux with height above the
  surface.
• The magnetic connectivity in the dynamic
  region between the photosphere and upper
  transition region is complex. This region is
  characterized by the presence of both twisted and
  non-twisted ß 1 horizontally-directed magnetic
  structures that thread the chromosphere and
  connect relatively distant concentrations of
  magnetic flux in the photosphere with the
  transition region footpoints of coronal
  structures.
• A non-ideal equation of state is necessary to
  obtain more realistic ratios of magnetic to gas
  pressure at, and just above, the model
  photosphere. However, an ideal treatment is
  capable of reproducing many of the global
  characteristics of the non-ideal atmospheres.

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