

Solar Surface MagnetoConvection
 Supergranule scale Convection
 Mesogranule scale MagnetoConvection
 Numerical Method
 Simulation Results
 Simulation vs. Observations

Oscillations
 Mode spectrum
 Mode frequencies
 Mode asymmetry
 TimeDistance Anaylsis
 Mode Driving and Damping

Radiation  Hydrodynamics
 Chromospheric dynamics and structure
 Dynamic line formation

Publications

Talks
Our goal is to understand convection in the solar
envelope: its role in transporting energy and angular momentum, in
generating the solar magnetic field, in providing energy to heat the
solar chromosphere and corona, and interacting with waves and
oscillations.
This goal is pursued by studying the results of realistic simulations
of magnetoconvection near the solar surface in collaboration with Åke
Nordlund (Copenhagen University), Mats Carlsson,
Viggo Hansteen and Boris Gudiksen (Oslo University), Junwei Zhao
(Stanford University) and Dali Georgobiani. [See
"Validation of TimeDistance Helioseismology by use of Realistic Simulations of Solar Convection", Astrophys. J., 659, 848857, (2007);
"Solar Small Scale
MagnetoConvection", Astrophys. J., 642, 12461255,
(2006);
"Observational manifestations
of solar magnetoconvection  centertolimb variation", Astrophys. J., 610, L137, (2004);
"Excitation
of Radial PModes in the Sun and Stars", Solar Phys.,
220, 229, (2004);
"Simulations of Solar Granulation:
I. General Properties'', Astrophys. J., 499, 914933,
(1998);
"Formation of Calcium H and K Grains", Astrophys. J., 481, 500, (1997)]
Supergranule Scale Convection
Movie (89 Mb) of Velocity and streamlines in vertical slice. Duration is 15
hours. Scale is 48 Mm wide x 20 Mm deep. Red are downflows and blue are
upflows. (courtesy Chris Henze, NASA)
At the top are the granules. Deeper down are larger structures.
Downflows are swept sideways and merged by the diverging upflows from below.
Some downflows are halted, beating there way down against the upflows.
Flow topology changes rapidly near the surface and very slowly at large depth.
Movie (136 Mb) shows the Finite Time Lyapunov Exponent Field, which closely
corresponds to the vorticity, for a time interval of 11.75 hours, in a
subdomain 21 Mm wide x 19 Mm high x 0.5 Mm thick, from a 48x48 Mm wide
by 20 Mm deep simulation. (Courtesy Bryan Green (AMTI/NASA).)
MesoGranular Scale MagnetoConvection
Movie (47 Mb) emergent intensity from magnetoconvection simulation showing
center to limb variation (courtesy Mats Carlsson).
Granules appear hilly when viewed toward the limb because they really are.
Hot granules emit their radiation higher up than cool intergranular lanes.
Movie (9 Mb) of Gband intensity at disk center and mu=0.6 from
magnetoconvection simulation (courtesy Mats Carlsson).
Small, strong magnetic field concentrations appear bright because their density is lower (to maintain pressure balance with their surroundings) so one sees
deeper into the Sun where it is hotter. Towards the limb, when one looks
through a low density, low opacity magnetic concentration one sees radiation
from the hot granule walls behind, which is called faculae.
Movie (15 Mb) of magnetic field being swept into the intergranular lanes by
the diverging upflows in granules and underlying larger flows.
Numerical Method
Solve the equations for conservation of mass, momentum and internal energy,
plus the induction equation for the magnetic field in conservative form,
on a staggered grid. Use 6th order finite difference spatial derivatives,
5th order interpolation, 3rd order RungeKutta time integration.
 Essential physics:

Equation of State includes ionization.
Ionization energy dominates internal energy near surface.

Radiative Transfer crucial
 Produces low entropy gas whose buoyancy work drives the
convection.
 Controls what we observe.
 Boundary of CZ occurs near tau=1
 Line blocking increases surface temperature

Diffusion
 stabilize numerics, damp short wavelength fluctuations.
 use hyperviscosity that enhances diffusivity at small scales and damps it at large scales.

Boundary Conditions
 Horizontal: periodic
 Top: Transmitting
 Bottom: set incoming fluid properties
 Bottom: node for vertical motions

Computational Domain
Convection Simulations
Mean Atmosphere
Topology controlled by mass conservation.
To conserve mass as it enters lower density layers, ascending fluid
diverges and turns over into a downflow in approximately a scale
height. Convective flow is like a fountain. It is consists of warm,
broad, fairly laminar, slow upflows, surrounded by cool, narrow,
turbulent, fast downdrafts. The horizontal size of the upflow cells
decreases as the as the scale height decreases approaching the
surface. Only a small fraction of ascending fluid actually reaches the
surface.
A granule. The top surface shows the emergent intensity. The interior
colors show the temperature and the arrows show the flow velocity.
A granule resembles a fountain. Hot fluid rises in the center and is
diverted horizontally into the intergranule lanes by excess pressure
over the granules. Fluid that reaches the surface cools, loses
entropy, and forms the cores of the downdrafts.
Convection is driven by radiative cooling
in a thin surface thermal boundary layer produces low entropy gas that
forms the cores of downdrafts.
Convection is inherently nonlocal. The
low entropy downdrafts, produced by radiative cooling in the surface
thermal boundary layer, generate most of the buoyancy work that drives
the convection: both large scale cellular flows and small scale
turbulent motions.
movie (127 Mb)
Entropy fluctuations and mean value.
Entropy fluctuation histogram.
Movie of entropy fluctuations in a downdraft (31 Mb)
(1.2 Mm)^{2} x 2.5 Mm deep
Convection characterized by turbulent
downdrafts and smooth upflows, not hierarchy of eddies.
Vorticity primarily in intergranular
lanes and downdrafts.
Movie of vorticity in same downdraft as entropy above (31 Mb)
(1.2 Mm)^{2} x 2.5 Mm deep
vorticity movie showing ring vortex and trailing vortex
tubes (127 Mb)
Horizontal scale increases
with increasing depth, also because of mass conservation.
No distinct mesogranular or supergranular cell size.
Vorticity slices
The top slice shows vortex tubes at the surface viewed from above. The
dark areas are free of significant vorticity and correspond to the hot,
upflowing, bright granules. The vortex tubes are concentrated in the
cool, downflowing, dark intergranule lanes. The twisting of these
vortices about one another shows the turbulent nature of the flow.
At increasing depth, lower slices, the vorticity becomes confined to the
boundary of the mesogranule and the large upflow cell in the interior of
the mesogranule is nearly vorticity free.
Velocity slices
Vertical velocity at the visible surface (top left) and 2, 4, 8, 12,
and 16 Mm below the surface. Each slice is 48 Mm square. Green and
blue are upflows, while yellow and red are downflows.
The downflows occur in more or less connected lanes
surrounding compact upflow cells  a granule at the surface and
continuously larger cells with increasing depth below the surface.
There are no special mesogranule or supergranule scales.
Simulation vs. Observations
 Granulation
(visible continuum).
Emergent Intensity
The emergent intensity in the simulation (top) is very similar to the
observed solar intensity (bottom) when the simulation results are
smoothed with a modulation transfer function to represent the effects
of the telescope and atmospheric seeing (middle). The observations are
from the Swedish Solar Observatory on La Palma, courtesy of the
Lockheed Palo Alto Research Center.
Size Spectrum
But beware: any image with sharp edged features (such as
) produces such a distribution.
Emergent intensity distribution
(visible continuum).
 Photospheric line profiles (visible)
(
"Line formation in solar granulation. I. Fe line shapes, shifts
and asymmetries", Asplund, M., Nordlund, Å., Trampedach,
R., Allende Prieto, C. and Stein, R. F., Astron. and Astroph.,
359, 729742, 2000)
Line Formation: Fe I+II lines
 LTE (use majority species, weak lines)
 Accurate wavelengths and gfvalues
 No free parameters (no micro
or macroturbulence, damping enhancement, etc)
Line widths, shifts, and shapes provide constraints
 Width <>
flow velocity (thermal speed is small)
 Shift <>
temperaturevelocity correlations
 Shape <>
details of convective overshoot, cf bisectors
Excellent agreement exists between simulated and
observed profiles of weak and intermediate strength FeI and FeII lines.
Without the convective and wave velocities, the line
profiles would differ drastically from the observed profiles:
Including velocities and sufficient resolution, produces close
agreement between simulated and observed profiles:
2D simulations do not give observed profiles
The average profile is a combination of profiles with very different
shifts, widths and shapes. (Thick red
line is average profile)
Line shapes depend on the details of the convection overshooting and
are revealed in the line bisectors.
Small differences exist between synthetic and observed profiles for
strong FeI lines. These can be used to improve
the physics of the upper photosphere (and chromosphere).
 Velocity Spectrum
The simulation velocity spectrum fits almost exactly with the
observed velocity spectrum from MDI and TRACE. This shows that our
simulations have the correct velocity amplitude and at large scales
the correct spectrum.
Agreement with observations gives confidence in model
of convection
 Mode Spectrum
komega diagram.
The simulations excite a rich spectrum of pmode oscillations (left),
very similar to the MDI diagram (right). The dark line is the theoretical
fmode.
 Mode Frequencies
(
"Convective contributions to the frequencies of solar oscillations",
Rosenthal, C. S., ChristensenDalsgaard, J., Nordlund, Å., Stein,
R. F., Trampedach, R., Astron. and Astrophys. 351, 689700, 1999)
Acoustic eigenmodes of the atmosphere in our simulation are excited. We
can study the properties of these simulation pmodes to understand
the solar pmodes better.
The mean atmosphere from the simulations gives pmode
eigenfrequencies in better agreement with observed modes than
standard, spherically symmetric, mixinglength models.
Standard, 1D, Spherically Symmetric, Mixing Length Model
3D Convection Simulation + Mixing Length Envelope Extension
High frequency modes' cavity is
enlarged by
 turbulent pressure support
(convergence to correct value ensured by line widths)
 3D radiative transfer effects
Don't see hot gas.
Average temperature higher for a given effective
temperature
Contribute equally to elevating photosphere by 150 km.
High frequency modes' frequency is reduced.
Remaining discrepancies in mode frequencies can now be used to
investigate details of the dynamical interaction of pmodes with
convection.
 Mode Asymmetry (helioseismology).
Pmode spectrum is asymmetric. Velocity power is larger toward
low frequencies and Intenstiy power is larger toward high
frequencies.
(
"Numerical Simulations of Oscillation Modes of the Solar Convection
Zone", Georgobiani, D. Kosovichev, A.G., Nigam, R., Nordlund,
Å, Stein, R.F. Astrophys. J., 530, L139L142, 2000)
IntensityVelocity phase jumps at modes.
 TimeDistance Diagram (helioseismology)
Time  Distance diagram.
The simulation and mdi timedistance diagrams are similar. Dark line is
the theoretical tddiagram. (courtesy Junwei Zhao)
 Pmode Excitation (helioseismology).
(
"Solar Oscillations and Convection: II. Excitation of Radial
Oscillations", Stein, R. F. and Nordlund, Å., Astrophys.
J., 546, 585603, 2001)
Stochastic, nonadiabatic, gas and turbulent pressure fluctuations
excite the pmodes via PdV work,
Mode excitation rate is,
Driving decreases at low frequencies
because the mode compression decreases and the mode mass
increases.
Driving decreases at high frequency
because the pressure fluctuations decrease.
Driving occurs closer to the surface at higher
frequencies
Logarithm (base 10) of the work integrand as a function of
frequency and depth (in unites of erg/cm^{2}/s).
Driving occurs in the intergranular lanes and
at the edges of granules.
Nonadiabatic pressure fluctuations at 100 km depth
in the 2  5 mHz range, with the contours of zero velocity at
the surface to outline the granules. The units of the pressure
fluctuations are 10^{3} dyne cm_{$2}. In this
frequency range, where the driving is maximal, the largest pressure
fluctuations occur at the edges of granules and inside the
intergranular lanes.
Mats Carlsson (Oslo University) and I have performed selfconsistent
nonLTE radiation hydrodynamics simulations of the propagation of
acoustic waves through the solar chromosphere. We find that the
chromosphere is dynamic and that static diagnostics do not accurately
represent its properties. For instance, enhanced
chromospheric emission, which corresponds to an outwardly increasing
semiempirical temperature structure, can be produced by wave motions
without any increase in the mean gas temperature. This is because in the
ultraviolet and visible the Planck function depends exponentially on the
temperature, so the emission tends to represent the temperature maxima
rather than the mean values. Thus, despite long
held beliefs, the sun may not have a classical chromosphere in magnetic
field free internetwork regions
(``Does a Non  Magnetic Solar Chromosphere Exist?'',
Carlsson & Stein, Ap. J. Lett., 440, L29 (1995)).
Chromospheric Temperature
Time averaged (Mean) gas temperature in the dynamical simulation and
the Semiempirical temperature that gives the best fit to the time
average of the intensity as a function of wavelength calculated from
the dynamical simulation.
Also shown are:
the minimum and maximimum (range) temperatures in the simulation,
the starting model temperature,
and the semiempirical model of Fontenala, Avrett & Loeser, 1993.
The semiempirical model giving the same intensities as
the dynamical simulation shows a chromospheric temperature rise while
the mean temperature in the simulation does not.
We have also simulated the generation of CaII H_{2V} bright
grains by acoustic shocks. The bright grains are produced by shocks
near 1 Mm above tau_{500}=1. The asymmetry of the line profile is
due to velocity gradients.
The formation of the CaII H_{2V} bright grains. The
contribution function to intensity (lower right) and the factors
entering its calculation, opacity and optical depth (upper left),
Source function (upper right) and tau*exp(tau) (lower left). The
functions are shown as greyscale images as functions of frequency in
the line (in velocity units) and height in the atmosphere. All
panels also show the velocity as a function of height (with upward
velocity positive to the left in the figure) and the height where
tau=1 (grey line). The top right panel, in addition, shows the
Planck function (dotted) and the source function (dashed), with high
values to the left. The source function (image, upper right panel)
is constant across the line at a given height, because of the
assumption of CRD, and is substantially below the Planck function
above 1 Mm due to the nonLTE decoupling. The bottom right panel
also shows the emergent intensity as a function of frequency.
The simulations closely match the observed
behavior of CaII H_{2V} bright grains down to the level of
individual grains
(
"Formation of Calcium H and K Grains",
Carlsson, M. and Stein, R. F., Astrophys. J. 481, 500, 1997).
The crucial lesson to learn from these studies is that the chromosphere is Dynamic and a static
analysis may not give correct results,
(
"The New Chromosphere", Carlsson, M. and Stein, R. F., in
New eyes to see inside the sun and stars : pushing the limits of
helio and asteroseismology with new observations from the ground and
from space, IAU Symposium 185, eds. F. L. Deubner, J.
ChristensenDalsgaard and D. Kurtz, (Kluwer:Dordrecht) 435446, 1998,
and
"Dynamic Behavior of the Solar Atmosphere", Stein, R. F. and
Carlsson, M., in Solar Convection and Oscillations and their
Relationship, eds. F.P.Pijpers, J. ChristensenDalsgaard, C.
Rosenthal, (Kluwer:Dordrecht), 261276, 1997).
This research is/was supported by NASA grants NAG NNG04GB92G, NAG 5
12450, NNX07AO71G, NNX07AH79G, NNX07AI08G, NNX08AH44G, and NSF grants
AST0205500 and AST0605738. This support is greatly appreciated and
has been instrumental in making progress on these projects. The
calculations were performed at: the NASA Advanced Supercomputing
facility, the National Center for Supercomputing Applications
(supported by NSF), UNIC Computing Center Copenhagen, and Michigan
State University Computer Center. Any opinions, findings, and
conclusions expressed in this material are those of the author(s)
and do not necessarily reflect the views of NASA or the NSF.
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Updated: 2008.05.02 (Friday) 15:02:11 EDT
Bob Stein's home page, email:
stein@pa.msu.edu