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Author name code: vigeesh
ADS astronomy entries on 2022-09-14
author:Vigeesh, Gangadharan
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Title: Acoustic-gravity wave propagation characteristics in 3D
radiation hydrodynamic simulations of the solar atmosphere
Authors: Fleck, Bernhard; Khomenko, Elena; Carlsson, Mats; Rempel,
Matthias; Steiner, Oskar; Riva, Fabio; Vigeesh, Gangadharan
2022cosp...44.2503F Altcode:
There has been tremendous progress in the degree of realism of
three-dimensional radiation magneto-hydrodynamic simulations of the
solar atmosphere in the past decades. Four of the most frequently
used numerical codes are Bifrost, CO5BOLD, MANCHA3D, and MURaM. Here
we test and compare the wave propagation characteristics in model
runs from these four codes by measuring the dispersion relation
of acoustic-gravity waves at various heights. We find considerable
differences between the various models.
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Title: Absorption of High-frequency Oscillations and Its Relation
to Emissivity Reduction
Authors: Waidele, Matthias; Roth, Markus; Vigeesh, Gangadharan;
Glogowski, Kolja
2021ApJ...913..108W Altcode: 2021arXiv210601745W
Sunspots are known to be strong absorbers of solar oscillation
modal power. The most convincing way to demonstrate this is done via
Fourier-Hankel decomposition (FHD), where the local oscillation field
is separated into in- and outgoing waves, showing the reduction in
power. Due to the Helioseismic and Magnetic Imager's high-cadence
Doppler measurements, power absorption can be investigated at
frequencies beyond the acoustic cutoff frequency. We perform an
FHD on five sunspot regions and two quiet-Sun control regions
and study the resulting absorption spectra α<SUB>ℓ</SUB>(ν),
specifically at frequencies ν > 5.3 mHz. We observe an unreported
high-frequency absorption feature, which only appears in the presence of
a sunspot. This feature is confined to phase speeds of one-skip waves
whose origins coincide with the sunspot's center, with v<SUB>ph</SUB>
= 85.7 km s<SUP>-1</SUP> in this case. By employing a fit to the
absorption spectra at a constant phase speed, we find that the peak
absorption strength ${\alpha }_{\max }$ lies between 0.166 and 0.222
at a noise level of about 0.009 (5%). The well-known absorption
along ridges at lower frequencies can reach up to ${\alpha }_{\max
}\approx 0.5$ . Thus our finding in the absorption spectrum is weaker,
but nevertheless significant. From first considerations regarding the
energy budget of high-frequency waves, this observation can likely be
explained by the reduction of emissivity within the sunspot. We derive
a simple relation between emissivity and absorption. We conclude that
sunspots yield a wave power absorption signature (for certain phase
speeds only), which may help in understanding the effect of strong
magnetic fields on convection and source excitation and potentially
in understanding the general sunspot subsurface structure.
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Title: On the effect of oscillatory phenomena on Stokes inversion
results
Authors: Keys, P. H.; Steiner, O.; Vigeesh, G.
2021RSPTA.37900182K Altcode: 2020arXiv200805539K
Stokes inversion codes are crucial in returning properties of the solar
atmosphere, such as temperature and magnetic field strength. However,
the success of such algorithms to return reliable values can be
hindered by the presence of oscillatory phenomena within magnetic
wave guides. Returning accurate parameters is crucial to both
magnetohydrodynamics (MHD) studies and solar physics in general. Here,
we employ a simulation featuring propagating MHD waves within a flux
tube with a known driver and atmospheric parameters. We invert the
Stokes profiles for the 6301 Å and 6302 Å line pair emergent from
the simulations using the well-known Stokes Inversions from Response
functions code to see if the atmospheric parameters can be returned
for typical spatial resolutions at ground-based observatories. The
inversions return synthetic spectra comparable to the original input
spectra, even with asymmetries introduced in the spectra from wave
propagation in the atmosphere. The output models from the inversions
match closely to the simulations in temperature, line-of-sight magnetic
field and line-of-sight velocity within typical formation heights of the
inverted lines. Deviations from the simulations are seen away from these
height regions. The inversions results are less accurate during passage
of the waves within the line formation region. The original wave period
could be recovered from the atmosphere output by the inversions, with
empirical mode decomposition performing better than the wavelet approach
in this task. <P />This article is part of the Theo Murphy meeting issue
`High-resolution wave dynamics in the lower solar atmosphere'.
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Title: Acoustic-gravity wave propagation characteristics in
three-dimensional radiation hydrodynamic simulations of the solar
atmosphere
Authors: Fleck, B.; Carlsson, M.; Khomenko, E.; Rempel, M.; Steiner,
O.; Vigeesh, G.
2021RSPTA.37900170F Altcode: 2020arXiv200705847F
There has been tremendous progress in the degree of realism of
three-dimensional radiation magneto-hydrodynamic simulations of the
solar atmosphere in the past decades. Four of the most frequently
used numerical codes are Bifrost, CO5BOLD, MANCHA3D and MURaM. Here
we test and compare the wave propagation characteristics in model
runs from these four codes by measuring the dispersion relation of
acoustic-gravity waves at various heights. We find considerable
differences between the various models. The height dependence of
wave power, in particular of high-frequency waves, varies by up to
two orders of magnitude between the models, and the phase difference
spectra of several models show unexpected features, including ±180°
phase jumps. <P />This article is part of the Theo Murphy meeting issue
`High-resolution wave dynamics in the lower solar atmosphere'.
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Title: On the influence of magnetic topology on the propagation of
internal gravity waves in the solar atmosphere
Authors: Vigeesh, G.; Roth, M.; Steiner, O.; Fleck, B.
2021RSPTA.37900177V Altcode: 2020arXiv201006926V
The solar surface is a continuous source of internal gravity waves
(IGWs). IGWs are believed to supply the bulk of the wave energy for
the lower solar atmosphere, but their existence and role for the energy
balance of the upper layers is still unclear, largely due to the lack
of knowledge about the influence of the Sun's magnetic fields on
their propagation. In this work, we look at naturally excited IGWs
in realistic models of the solar atmosphere and study the effect
of different magnetic field topographies on their propagation. We
carry out radiation-magnetohydrodynamic simulations of a magnetic
field free and two magnetic models-one with an initial, homogeneous,
vertical field of 100 G magnetic flux density and one with an initial
horizontal field of 100 G flux density. The propagation properties
of IGWs are studied by examining the phase-difference and coherence
spectra in the k<SUB>h</SUB> - ω diagnostic diagram. We find that IGWs
in the upper solar atmosphere show upward propagation in the model with
predominantly horizontal field similar to the model without magnetic
field. In contrast to that the model with predominantly vertical fields
show downward propagation. This crucial difference in the propagation
direction is also revealed in the difference in energy transported by
waves for heights below 0.8 Mm. Higher up, the propagation properties
show a peculiar behaviour, which require further study. Our analysis
suggests that IGWs may play a significant role in the heating of
the chromospheric layers of the internetwork region where horizontal
fields are thought to be prevalent. <P />This article is part of the
Theo Murphy meeting issue `High-resolution wave dynamics in the lower
solar atmosphere'.
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Title: Interaction of Magnetic Fields with a Vortex Tube at Solar
Subgranular Scale
Authors: Fischer, C. E.; Vigeesh, G.; Lindner, P.; Borrero, J. M.;
Calvo, F.; Steiner, O.
2020ApJ...903L..10F Altcode: 2020arXiv201005577F
Using high-resolution spectropolarimetric data recorded with the
Swedish 1 m Solar Telescope, we have identified several instances of
granular lanes traveling into granules. These are believed to be the
observational signature of underlying tubes of vortical flow with
their axis oriented parallel to the solar surface. Associated with
these horizontal vortex tubes, we detect in some cases a significant
signal in linear polarization, located at the trailing dark edge of
the granular lane. The linear polarization appears at a later stage of
the granular lane development, and is flanked by patches of circular
polarization. Stokes inversions show that the elongated patch of linear
polarization signal arises from the horizontal magnetic field aligned
with the granular lane. We analyze snapshots of a magnetohydrodynamic
numerical simulation and find cases in which the horizontal vortex
tube of the granular lane redistributes and transports the magnetic
field to the solar surface causing a polarimetric signature similar to
what is observed. We thus witness a mechanism capable of transporting
magnetic flux to the solar surface within granules. This mechanism is
probably an important component of the small-scale dynamo supposedly
acting at the solar surface and generating the quiet-Sun magnetic field.
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Title: Synthetic observations of internal gravity waves in the
solar atmosphere
Authors: Vigeesh, G.; Roth, M.
2020A&A...633A.140V Altcode: 2019arXiv191206435V
<BR /> Aims: We study the properties of internal gravity waves (IGWs)
detected in synthetic observations that are obtained from realistic
numerical simulation of the solar atmosphere. <BR /> Methods: We
used four different simulations of the solar magneto-convection
performed using the CO<SUP>5</SUP>BOLD code. A magnetic-field-free
model and three magnetic models were simulated. The latter three
models start with an initial vertical, homogeneous field of 10, 50,
and 100 G magnetic flux density, representing different regions of
the quiet solar surface. We used the NICOLE code to compute synthetic
spectral maps from all the simulated models for the two magnetically
insensitive neutral iron lines Fe I λλ 5434 Å and 5576 Å. We
carried out Fourier analyses of the intensity and Doppler velocities
to derive the power, phase, and coherence in the k<SUB>h</SUB> -
ω diagnostic diagram to study the properties of internal gravity
waves. <BR /> Results: We find the signatures of the internal gravity
waves in the synthetic spectra to be consistent with observations
of the real Sun. The effect of magnetic field on the wave spectra is
not as clearly discernible in synthetic observations as in the case of
numerical simulations. The phase differences obtained using the spectral
lines are significantly different from the phase differences in the
simulation. The phase coherency between two atmospheric layers in the
gravity wave regime is height dependent and is seen to decrease with
the travel distance between the observed layers. In the studied models,
the lower atmosphere shows a phase coherency above the significance
level for a height separation of ∼400 km, while in the chromospheric
layers it reduces to ∼100-200 km depending on the average magnetic
flux density. Conclusion. We conclude that the energy flux of IGWs
determined from the phase difference analysis may be overestimated
by an order of magnitude. Spectral lines that are weak and less
temperature sensitive may be better suited to detecting internal waves
and accurately determining their energy flux in the solar atmosphere.
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Title: Internal Gravity Waves in the Magnetized Solar
Atmosphere. II. Energy Transport
Authors: Vigeesh, G.; Roth, M.; Steiner, O.; Jackiewicz, J.
2019ApJ...872..166V Altcode: 2019arXiv190108871V
In this second paper of the series on internal gravity waves (IGWs),
we present a study of the generation and propagation of IGWs in a
model solar atmosphere with diverse magnetic conditions. A magnetic
field-free and three magnetic models that start with an initial,
vertical, homogeneous field of 10, 50, and 100 G magnetic flux density,
are simulated using the CO<SUP>5</SUP>BOLD code. We find that the
IGWs are generated in similar manner in all four models in spite of
the differences in the magnetic environment. The mechanical energy
carried by IGWs is significantly larger than that of the acoustic
waves in the lower part of the atmosphere, making them an important
component of the total wave energy budget. The mechanical energy flux
(10<SUP>6</SUP>-10<SUP>3</SUP> W m<SUP>-2</SUP>) is a few orders of
magnitude larger than the Poynting flux (10<SUP>3</SUP>-10<SUP>1</SUP>
W m<SUP>-2</SUP>). The Poynting fluxes show a downward component in
the frequency range corresponding to the IGWs, which confirm that
these waves do not propagate upward in the atmosphere when the fields
are predominantly vertical and strong. We conclude that, in the upper
photosphere, the propagation properties of IGWs depend on the average
magnetic field strength and therefore these waves can be potential
candidates for magnetic field diagnostics of these layers. However,
their subsequent coupling to Alfvénic waves is unlikely in a magnetic
environment permeated with predominantly vertical fields, and therefore
they may not directly or indirectly contribute to the heating of layers
above plasma-β less than 1.
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Title: On the effect of vorticity on the propagation of internal
gravity waves.
Authors: Vigeesh, G.; Steiner, O.; Calvo, F.; Roth, M.
2017MmSAI..88...54V Altcode:
We compare different models of solar surface convection to study
vorticity and how it can influence the propagation of internal
gravity waves. We conclude that simulations performed with higher grid
resolution may have a reduced gravity wave flux in the lower part of
the atmosphere due to strong vorticity. We also show that the vertical
extent of the allowed region of propagation depends on the magnetic
field inclination.
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Title: Internal Gravity Waves in the Magnetized Solar
Atmosphere. I. Magnetic Field Effects
Authors: Vigeesh, G.; Jackiewicz, J.; Steiner, O.
2017ApJ...835..148V Altcode: 2016arXiv161204729V
Observations of the solar atmosphere show that internal gravity
waves are generated by overshooting convection, but are suppressed
at locations of magnetic flux, which is thought to be the result of
mode conversion into magnetoacoustic waves. Here, we present a study
of the acoustic-gravity wave spectrum emerging from a realistic,
self-consistent simulation of solar (magneto)convection. A magnetic
field free, hydrodynamic simulation and a magnetohydrodynamic (MHD)
simulation with an initial, vertical, homogeneous field of 50 G flux
density were carried out and compared with each other to highlight the
effect of magnetic fields on the internal gravity wave propagation
in the Sun’s atmosphere. We find that the internal gravity waves
are absent or partially reflected back into the lower layers in the
presence of magnetic fields and argue that the suppression is due to
the coupling of internal gravity waves to slow magnetoacoustic waves
still within the high-β region of the upper photosphere. The conversion
to Alfvén waves is highly unlikely in our model because there is no
strongly inclined magnetic field present. We argue that the suppression
of internal waves observed within magnetic flux concentrations may also
be due to nonlinear breaking of internal waves due to vortex flows that
are ubiquitously present in the upper photosphere and the chromosphere.
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Title: CO5BOLD for MHD: progresses and deficiencies .
Authors: Steiner, O.; Calvo, F.; Salhab, R.; Vigeesh, G.
2017MmSAI..88...37S Altcode:
The magnetohydrodynamics module of CO5BOLD has been steadily improved
over the past decade and has been used for various solar and stellar
physical applications. We give an overview of current work with it
and of remaining and newly emerged shortcomings.
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Title: Gravity waves in magnetized solar atmospheres from MHD
simulations.
Authors: Jackiewicz, Jason; Vigeesh, Gangadharan
2014AAS...22412350J Altcode:
The solar atmosphere is believed to be a region where gravity waves are
generated and propagate, but a variety of effects makes observations of
them rather difficult. Measurements of gravity wave properties could,
however, show how they play an important role in the upper photosphere
and chromosphere and even deposit energy there. Here we show how
analysis of gravity waves from detailed numerical simulations can be
used to study magnetic fields and energy deposition in the atmosphere,
and how mode conversion to slow magneto-acoustic waves changes their
observable properties.
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Title: Seismology of Small-Scale Magnetic Features using Numerical
Simulation
Authors: Vigeesh, G.; Jackiewicz, J.
2013ASPC..478..259V Altcode:
We present results of 3D magnetohydrodynamic simulations as part of
a preliminary study aimed at understanding the interaction of seismic
waves with small-scale magnetic flux concentrations on the Sun. A model
solar atmosphere without magnetic fields (“Quiet Sun” model) and
a model with magnetic fields are constructed. We consider the solar
surface-gravity waves (f-mode) that are naturally excited in both the
simulated models and use them to measure travel times between different
locations on the surface. When compared to the field-free simulation,
we observe that a strong f-mode scattering occurs in the presence
of magnetic fields and results in travel-time shifts. With the help
of realistic numerical simulations, we show the seismic influence
of small-scale magnetic features in travel-time differences and its
possible effects on the helioseismic measurements of the structure
and dynamics of the solar interior.
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Title: Acoustic emission from magnetic flux tubes in the solar network
Authors: Vigeesh, G.; Hasan, S. S.
2013JPhCS.440a2045V Altcode: 2013arXiv1304.5193V
We present the results of three-dimensional numerical simulations to
investigate the excitation of waves in the magnetic network of the Sun
due to footpoint motions of a magnetic flux tube. We consider motions
that typically mimic granular buffeting and vortex flows and implement
them as driving motions at the base of the flux tube. The driving
motions generates various MHD modes within the flux tube and acoustic
waves in the ambient medium. The response of the upper atmosphere to
the underlying photospheric motion and the role of the flux tube in
channeling the waves is investigated. We compute the acoustic energy
flux in the various wave modes across different boundary layers
defined by the plasma and magnetic field parameters and examine the
observational implications for chromospheric and coronal heating.
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Title: First steps with CO5BOLD using HLLMHD and PP reconstruction .
Authors: Steiner, O.; Rajaguru, S. P.; Vigeesh, G.; Steffen, M.;
Schaffenberger, W.; Freytag, B.
2013MSAIS..24..100S Altcode:
We report on first experiences with real-life applications using
the MHD-module of CO5BOLD together with the piecewise parabolic
reconstruction scheme and present preliminary results of stellar
magnetic models with T<SUB>eff</SUB> = 4000 K to T<SUB>eff</SUB> =
5770 K.
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Title: Three-dimensional Simulations of Magnetohydrodynamic Waves
in Magnetized Solar Atmosphere
Authors: Vigeesh, G.; Fedun, V.; Hasan, S. S.; Erdélyi, R.
2012ApJ...755...18V Altcode: 2011arXiv1109.6471V
We present results of three-dimensional numerical simulations of
magnetohydrodynamic (MHD) wave propagation in a solar magnetic flux
tube. Our study aims at understanding the properties of a range of MHD
wave modes generated by different photospheric motions. We consider two
scenarios observed in the lower solar photosphere, namely, granular
buffeting and vortex-like motion, among the simplest mechanism for
the generation of waves within a strong, localized magnetic flux
concentration. We show that granular buffeting is likely to generate
stronger slow and fast magnetoacoustic waves as compared to swirly
motions. Correspondingly, the energy flux transported differs as a
result of the driving motions. We also demonstrate that the waves
generated by granular buffeting are likely to manifest in stronger
emission in the chromospheric network. We argue that different
mechanisms of wave generation are active during the evolution of a
magnetic element in the intergranular lane, resulting in temporally
varying emission at chromospheric heights.
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Title: Stokes Diagnostics of Magneto-Acoustic Wave Propagation in
the Magnetic Network on the Sun
Authors: Vigeesh, G.; Steiner, O.; Hasan, S. S.
2011SoPh..273...15V Altcode: 2011SoPh..tmp..349V; 2011arXiv1104.4069V
The solar atmosphere is magnetically structured and highly
dynamic. Owing to the dynamic nature of the regions in which the
magnetic structures exist, waves can be excited in them. Numerical
investigations of wave propagation in small-scale magnetic flux
concentrations in the magnetic network on the Sun have shown that
the nature of the excited modes depends on the value of plasma β
(the ratio of gas to magnetic pressure) where the driving motion
occurs. Considering that these waves should give rise to observable
characteristic signatures, we have attempted a study of synthesised
emergent spectra from numerical simulations of magneto-acoustic
wave propagation. We find that the signatures of wave propagation
in a magnetic element can be detected when the spatial resolution
is sufficiently high to clearly resolve it, enabling observations in
different regions within the flux concentration. The possibility to
probe various lines of sight around the flux concentration bears the
potential to reveal different modes of the magnetohydrodynamic waves
and mode conversion. We highlight the feasibility of using the Stokes-V
asymmetries as a diagnostic tool to study the wave propagation within
magnetic flux concentrations. These quantities can possibly be compared
with existing and new observations in order to place constraints on
different wave excitation mechanisms.
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Title: Wave propagation and energy transport in the magnetic network
of the Sun
Authors: Vigeesh, G.; Hasan, S. S.; Steiner, O.
2009A&A...508..951V Altcode: 2009arXiv0909.2325V
Aims. We investigate wave propagation and energy transport in
magnetic elements, which are representatives of small scale magnetic
flux concentrations in the magnetic network on the Sun. This is
a continuation of earlier work by Hasan et al. (2005, ApJ, 631,
1270). The new features in the present investigation include
a quantitative evaluation of the energy transport in the various
modes and for different field strengths, as well as the effect of the
boundary-layer thickness on wave propagation.<BR /> Methods: We carry
out 2D MHD numerical simulations of magnetic flux concentrations for
strong and moderate magnetic fields for which β (the ratio of gas to
magnetic pressure) on the tube axis at the photospheric base is 0.4 and
1.7, respectively. Waves are excited in the tube and ambient medium by
a transverse impulsive motion of the lower boundary.<BR /> Results: The
nature of the modes excited depends on the value of β. Mode conversion
occurs in the moderate field case when the fast mode crosses the β =
1 contour. In the strong field case the fast mode undergoes conversion
from predominantly magnetic to predominantly acoustic when waves are
leaking from the interior of the flux concentration to the ambient
medium. We also estimate the energy fluxes in the acoustic and magnetic
modes and find that in the strong field case, the vertically directed
acoustic wave fluxes reach spatially averaged, temporal maximum values
of a few times 10<SUP>6</SUP> erg cm<SUP>-2</SUP> s<SUP>-1</SUP> at
chromospheric height levels.<BR /> Conclusions: The main conclusions
of our work are twofold: firstly, for transverse, impulsive excitation,
flux tubes/sheets with strong fields are more efficient than those with
weak fields in providing acoustic flux to the chromosphere. However,
there is insufficient energy in the acoustic flux to balance the
chromospheric radiative losses in the network, even for the strong
field case. Secondly, the acoustic emission from the interface between
the flux concentration and the ambient medium decreases with the width
of the boundary layer.
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Title: Numerical simulation of wave propagation in magnetic network
Authors: Vigeesh, G.; Hasan, S. S.; Steiner, O.
2009IAUS..257..185V Altcode:
We present 2-D numerical simulations of wave propagation in the magnetic
network. The network is modelled as consisting of individual magnetic
flux sheets located in intergranular lanes. They have a typical
horizontal size of about 150 km at the base of the photosphere and
expand upward and become uniform. We consider flux sheets of different
field strengths. Waves are excited by means of transverse motions at
the lower boundary, to simulate the effect of granular buffeting. We
look at the magneto-acoustic waves generated within the flux sheet
and the acoustic waves generated in the ambient medium due to the
excitation. We calculate the wave energy fluxes separating them into
contributions from the acoustic and the Poynting part and study the
effect of the different field strengths.
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Title: Numerical simulation of wave propagation in the presence of
a magnetic flux sheet
Authors: Vigeesh, G.; Steiner, O.; Hasan, S. S.
2008ESPM...12.3.24V Altcode:
We model network magnetic fields as consisting of individual magnetic
flux sheets located in intergranular lanes. With a typical horizontal
size of about 150 km at the base of the photosphere, they expand upward
and merge with their neighbors at a height of about 600 km. Above
a height of approximately 1000 km the magnetic field starts to
become uniform. Granular buffeting is thought to excite waves in this
medium, which is modeled by means of transversal motions at the lower
boundary. The transverse driving, generates both fast and slow waves
within the flux sheet and acoustic waves in the ambient medium. We
consider flux sheets of different field strengths and different
boundary-layer widths. Separating the energy flux of the waves into
contributions due to the acoustic flux and the Poynting flux, we show
the longitudinal and transversal components of both and study their
temporal evolution.
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Title: First local helioseismic experiments with CO<SUP>5</SUP>BOLD
Authors: Steiner, O.; Vigeesh, G.; Krieger, L.; Wedemeyer-Böhm, S.;
Schaffenberger, W.; Freytag, B.
2007AN....328..323S Altcode: 2007astro.ph..1029S
With numerical experiments we explore the feasibility of using high
frequency waves for probing the magnetic fields in the photosphere and
the chromosphere of the Sun. We track a plane-parallel, monochromatic
wave that propagates through a non-stationary, realistic atmosphere,
from the convection-zone through the photosphere into the magnetically
dominated chromosphere, where it gets refracted and reflected. We
compare the wave travel time between two fixed geometrical height levels
in the atmosphere (representing the formation height of two spectral
lines) with the topography of the surface of equal magnetic and thermal
energy density (the magnetic canopy or β=1 contour) and find good
correspondence between the two. We conclude that high frequency waves
indeed bear information on the topography of the `magnetic canopy'.
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Title: Wave Propagation in the Magnetic Network on the Sun
Authors: Hasan, S. S.; Vigeesh, G.; van Ballegooijen, A. A.
2006IAUS..233..116H Altcode:
Hasan et al. (2005) have recently presented 2-D dynamical calculations
on wave propagation in in the magnetic network of the Sun. The latter
is idealized as consisting of non-potential flux tubes in the quiet
solar chromosphere. It is of interest to understand how the nature of
wave propagation is influenced by the choice of initial equilibrium
configuration of the magnetic field. We examine this by comparing
the earlier calculations with those when the network is modelled as
a potential structure. Our calculations demonstrate that the nature
of the wave propagation is significantly different, particularly
the transport of energy which for the potential case, occurs more
isotropically than for the non-potential configuration.
