Issue |
A&A
Volume 515, June 2010
|
|
---|---|---|
Article Number | A107 | |
Number of page(s) | 5 | |
Section | The Sun | |
DOI | https://doi.org/10.1051/0004-6361/200913846 | |
Published online | 15 June 2010 |
A photospheric bright point model
S. Shelyag - M. Mathioudakis - F. P. Keenan - D. B. Jess
Astrophysics Research Centre, School of Mathematics and Physics, Queen's University, Belfast, BT7 1NN, Northern Ireland, UK
Received 10 December 2009 / Accepted 8 March 2010
Abstract
Aims. A magneto-hydrostatic model is constructed with
spectropolarimetric properties close to those of solar photospheric
magnetic bright points.
Methods. Results of solar radiative magneto-convection
simulations are used to produce the spatial structure of the vertical
component of the magnetic field. The horizontal component of magnetic
field is reconstructed using the self-similarity condition, while the
magneto-hydrostatic equilibrium condition is applied to the standard
photospheric model with the magnetic field embedded. Partial ionisation
processes are found to be necessary for reconstructing the correct
temperature structure of the model.
Results. The structures obtained are in good agreement with
observational data. By combining the realistic structure of the
magnetic field with the temperature structure of the quiet solar
photosphere, the continuum formation level above the equipartition
layer can be found. Preliminary results are shown of wave propagation
through this magnetic structure. The observational consequences of the
oscillations are examined in continuum intensity and in the Fe I
6302 Å magnetically sensitive line.
Key words: Sun: oscillations - Sun: photosphere - Sun: surface magnetism - plasmas - magnetohydrodynamics (MHD) - radiative transfer
1 Introduction
The study of magnetic elements at very small scales is one of the most important topics in solar physics. Magnetic bright points (MBPs) are ubiquitous in the solar photosphere. They have small diameters, typically less than 300 km, and are found in the intergranular lanes. MBPs correspond to areas of kilogauss fields, are best observed in G-band disk centre images and are numerous in active regions or near sunspots. They are formed by a complex process involving the interaction of the magnetic field with the convectively unstable hot plasma. The physical processes associated with their formation have been outlined in Schüssler et al. (2003); Carlsson et al. (2004); Shelyag et al. (2004), using forward modelling of radiative magneto-convection in the solar photosphere and upper convection zone.
Despite the overall success of photospheric and sub-photospheric radiative magneto-convective models to reproduce many of the observational properties of solar radiation, we still do not fully understand the physical processes involved in the strongly magnetised photospheric plasma. In particular, it is difficult to use the results of these simulations for studies of acoustic wave propagation through the solar atmosphere and interior. Strong convective motions of the photospheric plasma can hide the signatures of acoustic waves, making them a difficult subject in both numerical and observational investigations.
The development of new methods for inferring the properties of solar plasma using sound waves
have been followed by the successful modelling of the magneto-acoustic properties in the solar
atmosphere and interior (Hanasoge et al. 2007; Shelyag et al. 2009; Steiner 2009; Parchevsky & Kosovichev 2009; Khomenko et al. 2009; Hasan & van Ballegooijen 2008; Vigeesh et al. 2009; Fedun et al. 2009; Hasan et al. 2005).
However, due to the non-locality of radiative processes in the solar
atmosphere, a direct comparison of the plasma parameters at a certain geometrical depth in the computational box with the solar radiation parameters at a given optical depth may not be entirely correct. The non-locality of radiative transport must be taken into account. Khomenko & Collados (2009)
have recently suggested that the changes in the height of continuum
formation with respect to the equipartition layer (the layer where the
Alfvén speed is equal to the sound speed,
), may help to explain the appearance of the high-frequency acoustic haloes around sunspots.
Numerical simulations of solar wave phenomena require a static magnetic configuration model which incorporates as many physical properties of the real Sun as possible. In this paper we provide a recipe to create such a model, based on the results of numerical modelling of magneto-convection in the photosphere. We demonstrate that the spectropolarimetric properties of the magnetic configuration we created successfully reproduces those of MBPs. In Sect. 2 we describe the technique used to reconstruct the magnetic and thermal parameters of the MBP model. The results of the spectropolarimetric simulations using the model are presented in Sect. 3. In Sect. 4 we show our preliminary results on the wave mode conversion in the MBP and discuss the possible observational signatures. Our concluding remarks are presented in Sect. 5.
2 Static bright point model
We use a snapshot from the ``plage'' magneto-convection simulation of the solar photosphere undertaken with the MURaM code (Vögler et al. 2005) to produce the average MBP model. Since the average
magnetic field flux in this snapshot is relatively high
,
a large number of
intergranular magnetic field concentrations are generated. The magnetic field concentrations appear
bright in the continuum
,
and in the G band. This snapshot has been
used to demonstrate the magnetic nature of the G-band bright points (Schüssler et al. 2003; Shelyag et al. 2004).
To reveal the basic structure of the vertical component of the magnetic field B0z
in the MBPs, we average the depth dependences of the vertical magnetic
field strength over the bright points, which are selected by their
enhanced G-band intensity and magnetic field strength. Figure 1 shows B0z as a function of depth. Note that z=0 on the depth scale corresponds to the average optical depth
for the quiet Sun, and thus does not include Wilson depression (Wilson & Maskelyne 1774). The Wilson depression is about 100 km, as will be shown later.
![]() |
Figure 1:
Vertical magnetic field component B0z as a function of depth for the photospheric MBPs. A depth level of z=0 corresponds to
|
Open with DEXTER |
We use the self-similarity assumption (Shelyag et al. 2009; Schüssler & Rempel 2005; Schlüter & Temesváry 1958; Gordovskyy & Jain 2007) to reconstruct the horizontal component of the magnetic field. Below we briefly discuss its governing equations.
For the prescribed vertical magnetic field component B0z(z), defined along the axis of the symmetric magnetic field configuration, the (divergence-free) two-dimensional magnetic field
can be defined as:
and
where f is an arbitrary function which describes how the vertical component of the field expands with height.
For the simulations, we choose the following grid parameters of the
numerical domain in which the magnetic field is embedded: the vertical
and horizontal extents of the domain are
and
,
respectively, which are resolved in
grid cells.
In Fig. 2 we show the magnetic field configuration calculated for this domain using the above definition. The function f
takes zero values on the side boundaries of the domain and is chosen in
such a way that the radius of the magnetic field region is
approximately
at the height corresponding to the quiet Sun continuum formation level.
![]() |
Figure 2: Vertical ( left) and horizontal ( right) components of the reconstructed magnetic field for the MBP model. |
Open with DEXTER |
The magnetic field changes the thermodynamic parameters of the plasma where it is embedded. These
changes can be quite significant for the plasma dynamics and wave propagation (see e.g. Shelyag et al. 2009).
To construct the magneto-hydrostatic configuration, we substitute the
values of the magnetic field components obtained using Eqs. (1) and (2) into the equation of magneto-hydrostatic equilibrium
Here the magnetic field strength is normalised by the factor

We use the standard model of the solar photosphere (Spruit 1974)
as the unperturbed background model. The unperturbed pressure is
recalculated from the standard density profile using the hydrostatic
equilibrium condition
.
Then the perturbations to the pressure p and density
,
obtained from the solution of Eq. (3), are added to the unperturbed density and pressure dependencies.
Partial ionisation effects are taken into account when the plasma
internal energy and temperature are calculated. The system of Saha
equations is solved for the eleven most abundant elements in the solar
photosphere to produce the tabulated functions of internal energy
and temperature
.
Values of internal energy and temperature are then obtained by
interpolation. The density, pressure, internal energy and temperature
structures are shown in Fig. 3,
where the lower right panel shows a significant temperature increase
above the MBP. This temperature increase is caused by the increased
magnetic tension in the upper layers. A similar, but somewhat weaker,
temperature increase is observed in the dynamic radiative
magneto-convection simulations.
![]() |
Figure 3:
The theoretical density ( top left), pressure ( top right), internal energy per unit volume ( bottom left) and temperature ( bottom right) structures of the MBP. In the bottom right panel, the white line shows the 4300 Å continuum formation level
|
Open with DEXTER |
For a direct comparison of the radiative properties of the model with
observations, we need to know the position of the continuum formation
at some wavelength relative to the equipartition layer
.
The thermal and magnetic structure of the average MBP is shown in the lower right panel of Fig. 3, along with the equipartition layer and continuum formation level
.
The latter curve demonstrates the presence of Wilson depression of the
order of 100 km in the magnetised region, similar to that observed
for sunspots. The Wilson depression in the MBP model is significantly
less than that of sunspots (Solanki 1993; Watson et al. 2009).
The equipartition layer, as is evident from the figure, is located
deeper than the continuum formation layer in the region of the
strongest magnetic field. This means that in the MBP centre, the
4300 Å continuum is originating from strongly magnetised
plasma with
and
,
and a variety of MHD effects, including the magneto-acoustic mode conversion, may be observable at this wavelength.
3 Radiative diagnostics
The line profile simulation code STOPRO (Berdyugina et al. 2003; Solanki 1987; Shelyag et al. 2007a; Frutiger 2000; Shelyag et al. 2004)
is used to perform the line profile calculations. This code is designed
to compute wavelength-dependent intensities and normalised full Stokes
vectors for the atomic and molecular line profiles in the LTE
approximation.We use STOPRO to calculate the continuum intensities at
4300 Å and 5000 Å, as well as G-band intensities for
each of the 200 vertical rays of the MBP model described in
Sect. 2. To calculate these we employ a procedure similar to that
described in detail in Shelyag et al. (2004).
The G-band intensities are derived by convolving the simulated spectrum
with a 10 Å filter, centered at 4305 Å, and integrating
this over the range 4295 Å-4315 Å. The intensities are shown
in Fig. 4, and have been
normalised by the values taken in the non-magnetic part of the domain
corresponding to granulation. As is evident from the figure, the MBP
model appears to be bright in its middle x=0. Darkenings, corresponding to the integranular lanes surrounding the MBPs, are found at
and
.
![]() |
Figure 4:
G-band intensities for the average MBP model. The darkenings, corresponding to the integranular lanes, are found at
|
Open with DEXTER |
![]() |
Figure 5:
Stokes-I (top) and -V (bottom) profiles calculated for magnetic flux concentration at x=0, (left), and
|
Open with DEXTER |
The 6302.49 Å Fe I transition is often used as a diagnostic for the photospheric magnetic field, and the Stokes-I and -V profiles for this line are shown in Fig. 5 for two positions in the model. The I profile is split by strong magnetic field in the centre of the MBP, but the depth of the profile is reduced due to a smaller temperature gradient. In the weaker magnetic field (Fig. 5, right column), the absorption is stronger, and the Stokes-V profile has a larger amplitude. However, the wavelength separation between the left and right lobes of Stokes-V remain approximately the same in both cases. It should be pointed out that the observational profiles are also influenced by the Doppler shift, caused by the bulk motions of the convecting plasma. Our model does not account for convective velocities and thus cannot reproduce the exact central wavelength and C-shape of the observed absorption line profiles.
4 Wave propagation in the MBP
We use the code SAC to perform simulations of wave propagation and mode conversion in the MBP
(Fedun et al. 2009; Shelyag et al. 2009).
The code solves the full ideal MHD equations in a two- or
three-dimensional Cartesian grid, and has previously been used for a
variety of problems in the physics of solar oscillations. The boundary
conditions implemented in the code are ``transparent'' conditions of
the simplest type. These conditions lead to an artificial reflection
from the boundary layers, however, for well-resolved perturbations the
amplitudes of reflected waves are quite small. The full description of
the methods used, boundary conditions and tests are presented in detail
by Shelyag et al. (2008). For the
simulations, the code has been modified to incorporate the equation of
state and account for partial ionisation processes. At each time step,
the local internal energy- and density-dependent adiabatic index
is calculated using the pre-calculated tabulated
function
.
The adiabatic index is then used in the equation of state to relate the
kinetic pressure to the internal energy values, thus closing the system
of MHD equations.
Oscillations are generated using a single temporally continuous
acoustic source located in the non-magnetic part of the numerical
domain. This source is coherent, acts in the horizontal direction vx, extents vertically over the whole height of the domain, and its period is
.
The amplitude of the source is
,
is constant over the domain depth and is chosen to be sufficiently
small to keep the oscillations in the linear regime. The diffusive
damping introduced into the code by increasing the hyperdiffusion
coefficients to 0.2, combined with the low source amplitude, prevent
the convectively unstable domain from going into the convective regime
for a sufficiently long time.
A snapshot of the vertical velocity component, taken at
from the start of the simulation, is shown in Fig. 6. An oscillatory pattern, bounded by the equipartition level
(marked
by the green line in the figure), is easily distinguishable. This
pattern is produced by an interference of the slow and fast
magneto-acoustic waves, propagating differently in the region. Most of
the velocity pattern is above the continuum formation level
(white dashed line), and should therefore affect the absorption line profiles.
![]() |
Figure 6:
Vertical velocity component in the domain. Lines as in lower right panel of Fig. 3.
A complicated velocity pattern, produced by the interaction of the
oncoming wave with the magnetic field concentration, is visible above
the equipartition layer
|
Open with DEXTER |
Figure 7 shows the 5000 Å continuum intensity variations in non-magnetic media outside the MBP (dashed lines) and in the centre of the MBP (solid lines). It is evident from the figure that both the absolute intensity variation (upper plot) and the intensity variation normalised to the average continuum intensity (lower plot), are larger in the MBP than in non-magnetic media. This is caused by the larger continuum intensity and stronger variation of temperature in the deeper (due to depression) layers of the solar photosphere in the MBP. Although the source amplitude is sufficiently low to keep the linear character of the intensity oscillations in the non-magnetic part of the domain (dashed lines), a non-linear character of the intensity oscillations in the MBP is also observed (solid lines). The steepening of the wave front observed in the figure is caused by conversion of the linear wave into the non-linear regime in the partially evacuated plasma of the MBP.
![]() |
Figure 7:
Absolute (upper plot) and relative ( lower plot) continuum intensity oscillations in the non-magnetic media (
|
Open with DEXTER |
![]() |
Figure 8: Filtered Stokes-V intensities, calculated for the simulated time series. |
Open with DEXTER |
An oscillatory behaviour can also be detected in the magnetic field.
To reveal the observational consequences of the oscillations in the
magnetic field, we apply the standard observational technique of
measuring the Stokes-V profile. We choose a narrow bandpass filter with a width of
,
centered
from the Fe I line centre. We then apply this filter to the Stokes-V profiles data, calculated for the whole x-t domain. The resultant Stokes-V intensities are shown in Fig. 8.
The vertical component of the magnetic field in the region of the line
formation is proportional to the intensity of the Stokes-V filter. However, it is worth noting that the Stokes-V profile for the 6302 Å line saturates in regions of strong magnetic field. As Fig. 8 demonstrates, the intensity at x=0 is equal to the intensity at about
.
Also, the magnetic field strength values, restored from the Stokes-V
intensities at these positions, are virtually indistinguishable, thus
making the 6302.49 Å Fe I line unsuitable for measurements of
strong magnetic fields. The variation in the filtered Stokes V
amplitudes in the centre of MBP is about 25-30% in our simulations.
This value is quite large and should therefore be detectable by
observations.
The magnetic field configuration in the simulations is
localised and non-uniform, and the line-of-sight therefore crosses
regions of both zero and non-zero magnetic field. The initial static
model does not produce an asymmetry in the Stokes-V profiles, since it has no velocities which are necessary to produce an asymmetry (see e.g. Sankarasubramanian & Rimmele 2002; Bellot Rubio et al. 2000).
In the presence of a flow, caused by the perturbation introduced into
the numerical box, this magnetic field configuration can lead to
asymmetric Stokes-V profiles (Solanki 1993). The simplest measure of the area asymmetry is the integral of the Stokes-V profile over the wavelength
.
The V profile area asymmetry map is shown in Fig. 9. The normalised amplitude of Stokes-V area asymmetry is of the order of 2% in the center of the MBP. This value is close to the observed
values of Stokes-V area asymmetries, which are about 3% (Bellot Rubio et al. 2000). Thus, the variation of the Stokes-V area asymmetry produced by the perturbation is large enough to be detected by modern instruments.
![]() |
Figure 9: Stokes-V asymmetries, calculated for the simulated time series. |
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5 Concluding remarks
We have presented the recipe for creating a static numerical model of a photospheric MBP. This model allows the study of oscillatory properties of such small magnetic configurations in the solar photosphere directly by measuring their radiation intensity and polarimetric properties. The magnetic field is extracted from dynamic simulations of solar radiative magneto-convection and reconstructed using the assumption of self-similarity. Thus, our static model inherits many observational properties of the full dynamic simulation.
The model reproduces the G-band brightening in the MBP centre. Stokes-I and -V profiles for the
FeI
line are comparable to solar observations in both the magnetised region
and in the region corresponding to the non-magnetic solar granulation.
The continuum formation level in the centre of the MBP is located in
the region of strongly magnetised plasma, where the Alfvén speed is
greater than that of sound.
We have used the model to examine the observational
consequences of sound wave propagation though the magnetic field
concentration of the MBP. This preliminary investigation shows that the
variation of continuum intensity is more pronounced in the MBP compared
to the average granule. Using the radiative diagnostics with the Stokes-V profile of the
Fe I absorption line, we demonstrate the detectability of the magnetic
field variation in the bright point and show the appearance of the
Stokes-V profile asymmetry caused by the oscillations.
The radiative heating term is not included into the system of MHD equations used to perform the wave dynamics study. The absence of the heating term may result in reduced vertical radiative flux in the magnetic flux tube. As a result, we have somewhat smaller continuum and G-band intensity in the MBPs, compared to the full simulations and the real Sun. However, for wave dynamics the effect of the radiative term is expected to be small. As it has recently been shown by Yelles Chaouche et al. (2009), the magnetic flux tubes in the solar photosphere are ``reasonably well reproduced'' by a thin flux tube approximation with no radiative term included.
Future investigations will examine the response to sources at different locations and with a range of frequencies. The behaviour and observational signatures of Alfvén waves in small magnetic elements is particularly important when used as a diagnostic for solar plasma parameters, and for understanding the energy transport to the corona. Thus we plan to extend the model to three dimensions.
AcknowledgementsThis work has been supported by the UK Science and Technology Facilities Council (STFC). F.P.K. is grateful to AWE Aldermaston for the award of a William Penney Fellowship.
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All Figures
![]() |
Figure 1:
Vertical magnetic field component B0z as a function of depth for the photospheric MBPs. A depth level of z=0 corresponds to
|
Open with DEXTER | |
In the text |
![]() |
Figure 2: Vertical ( left) and horizontal ( right) components of the reconstructed magnetic field for the MBP model. |
Open with DEXTER | |
In the text |
![]() |
Figure 3:
The theoretical density ( top left), pressure ( top right), internal energy per unit volume ( bottom left) and temperature ( bottom right) structures of the MBP. In the bottom right panel, the white line shows the 4300 Å continuum formation level
|
Open with DEXTER | |
In the text |
![]() |
Figure 4:
G-band intensities for the average MBP model. The darkenings, corresponding to the integranular lanes, are found at
|
Open with DEXTER | |
In the text |
![]() |
Figure 5:
Stokes-I (top) and -V (bottom) profiles calculated for magnetic flux concentration at x=0, (left), and
|
Open with DEXTER | |
In the text |
![]() |
Figure 6:
Vertical velocity component in the domain. Lines as in lower right panel of Fig. 3.
A complicated velocity pattern, produced by the interaction of the
oncoming wave with the magnetic field concentration, is visible above
the equipartition layer
|
Open with DEXTER | |
In the text |
![]() |
Figure 7:
Absolute (upper plot) and relative ( lower plot) continuum intensity oscillations in the non-magnetic media (
|
Open with DEXTER | |
In the text |
![]() |
Figure 8: Filtered Stokes-V intensities, calculated for the simulated time series. |
Open with DEXTER | |
In the text |
![]() |
Figure 9: Stokes-V asymmetries, calculated for the simulated time series. |
Open with DEXTER | |
In the text |
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