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( a ) The three-layer model, where there is an inhomogeneous and stratified magnetised solar atmosphere with a steady sub-photospheric solar interior and ( b ) the resonantly coupled global p modes and their zoom in ( c ) in this three-layer solar model (after Pintér et al. 2008)
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Helioseismology is practically the only efficient experimental way of probing the solar interior. Without it, the results
of theoretical solar models would remain untested and, consequently, less reliable when applying them for investigating remote
stars. Hence, having a firm understanding of the applicability and reliability of helioseismology and...
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Citations
... Theoretical investigations, on the other hand, in global helioseismology, have been carried out extensively over several decades to predict the effect of magnetic fields on helioseismic waves (see, e.g., Nye & Thomas 1976;Adam 1977;Thomas 1983;Campos 1983;Lee & Roberts 1986;Miles & Roberts 1992;Jain & Roberts 1991;Cally 2007;Foullon & Roberts 2005;Pintér & Erdélyi 2011;Campos & Marta 2015;Pintér 2015). It is well established that magnetic effects upshift the frequencies of acoustic modes of stellar oscillations (Gough & Thompson 1990;Goldreich et al. 1991;Dziembowski & Goode 2004). ...
We solve for waves in an isothermal, stratified medium with a magnetic field that points along a direction perpendicular to that of gravity and varies exponentially in the direction of gravity. We find exact analytical solutions for two different cases: (a) waves propagating along the direction of the magnetic field and (b) waves propagating along the direction of the gravity. In each case, we find solutions in terms of either the hypergeometric functions or their confluent cousins. We solve the resultant transcendental dispersion relation numerically. The eigenfrequencies decrease with increasing degree of the spatial inhomogeneity of the magnetic field. Further, the nodes of the eigenfunctions leak toward regions of lower Alfvén wave speed due to softened wave-reflection in such regions. Such changes in the dispersion relation and the mode structures may allow the detection of magnetic fields buried in the stellar interior.
... For review see e.g. Christensen-Dalsgaard (2002); Erdélyi (2006b,a); Thompson (2006); Pintér and Erdélyi (2011). ...
... There is a significant number of works reporting on observational, theoretical and computational studies of p-mode phenomena see e.g. Christensen-Dalsgaard (2002); Erdélyi (2006b,a); Thompson (2006); Pintér and Erdélyi (2011). Observational and theoretical analysis generally describes mechanisms for the propagation of energy into the chromosphere, tansition region and into the solar corona in magnetic structures (e.g. ...
The solar atmosphere exhibits a diverse range of wave phenomena, where one of the earliest discovered was the five-minute global acoustic oscillation, also referred to as the p-mode. The analysis of wave propagation in the solar atmosphere may be used as a diagnostic tool to estimate accurately the physical characteristics of the Sun’s atmospheric layers.
In this paper, we investigate the dynamics and upward propagation of waves which are generated by the solar global eigenmodes. We report on a series of hydrodynamic simulations of a realistically stratified model of the solar atmosphere representing its lower region from the photosphere to low corona. With the objective of modelling atmospheric perturbations, propagating from the photosphere into the chromosphere, transition region and low corona, generated by the photospheric global oscillations the simulations use photospheric drivers mimicking the solar p-modes. The drivers are spatially structured harmonics across the computational box parallel to the solar surface. The drivers perturb the atmosphere at 0.5 Mm above the bottom boundary of the model and are placed coincident with the location of the temperature minimum. A combination of the VALIIIC and McWhirter solar atmospheres are used as the background equilibrium model.
We report how synthetic photospheric oscillations may manifest in a magnetic field free model of the quiet Sun. To carry out the simulations, we employed the magnetohydrodynamics code, SMAUG (Sheffield MHD Accelerated Using GPUs).
Our results show that the amount of energy propagating into the solar atmosphere is consistent with a model of solar global oscillations described by Taroyan and Erdélyi (2008) using the Klein-Gordon equation. The computed results indicate a power law which is compared to observations reported by Ireland et al. (2015) using data from the Solar Dynamics Observatory/Atmospheric Imaging Assembly.
... Sunspots and pores are just two of these many structures, and they are known to display solar global oscillations. See a recent review by, e.g., Pintér & Erdélyi (2011). ...
... These oscillations are seen in intensity, line-of-sight (LOS) velocity, and LOS magnetic field. The source of the five-minute oscillation is a result of forcing by the five-minute (p-mode) global solar oscillation (Marsh & Walsh 2008), which forms the basis of helioseismology (Thompson 2006;Pintér & Erdélyi 2011). The five-minute oscillations are typically seen in simple molecular and non-ionized metal lines, which form low in the umbral photosphere and are moderately suppressed not only in the penumbra, but also in the chromospheric atmosphere above the umbra (Bogdan & Judge 2006). ...
By focusing on the oscillations of the cross-sectional area and the intensity of magnetic waveguides located in the lower solar atmosphere, we aim to detect and identify magnetohydrodynamic (MHD) sausage waves. Capturing several series of high-resolution images of pores and sunspots and employing wavelet analysis in conjunction with empirical mode decomposition (EMD) makes the MHD wave analysis possible. For this paper, two sunspots and one pore (with a light bridge) were chosen as representative examples of MHD waveguides in the lower solar atmosphere. The sunspots and pore display a range of periods from 4 to 65 minutes. The sunspots support longer periods than the pore - generally enabling a doubling or quadrupling of the maximum pore oscillatory period. All of these structures display area oscillations indicative of MHD sausage modes and in-phase behaviour between the area and intensity, presenting mounting evidence for the presence of the slow sausage mode within these waveguides. The presence of fast and slow MHD sausage waves has been detected in three different magnetic waveguides in the lower solar photosphere. Furthermore, these oscillations are potentially standing harmonics supported in the waveguides which are sandwiched vertically between the temperature minimum in the lower solar atmosphere and the transition region. Standing harmonic oscillations, by means of solar magneto-seismology, may allow insight into the sub-resolution structure of photospheric MHD waveguides.
... The branch of solar physics that exploits the observed solar surface oscillations to understand the properties of the interior is called helioseismology, seismic investigation of the Sun. Spherical harmonics can be used to mathematically describe small-amplitude (linear) oscillations in a spherical, non-rotating medium with adiabatic stratification, that models the solar interior with no atmosphere (see details of such models in Christensen-Dalsgaard (1980), Campbell & Roberts (1989), and Pintér & Erdélyi (2011)). The frequencies of the possible oscillations are ...
... In another atmospheric profile, which represents the presence of a strong magnetic field, B 0 (z) is kept constant. The mathematical description of analytical and numerical solutions of the governing equations for the perturbed quantities, equations (12), including short-and long-wavelength approximations, in atmospheres with constant Alfvén speed and also with constant magnetic field strength, are given in detail in Erdélyi (2006a, b) and Pintér & Erdélyi (2011). ...
Both the interior and the atmosphere of the Sun give a wide range of oscillations and waves. Interactions between waves and their highly structured and dynamic environment strongly influence the various properties of the waves. Understanding those possible interactions could provide priceless diagnostic tools in the search for hidden aspects of the solar interior and atmosphere. This article is an attempt to overview briefly our current understanding of how global helioseismic oscillations, f and p acoustic waves, interact with plasma flows and magnetic fields in the solar atmosphere.
... However, the p-modes can be modified by the magnetic field (for reviews with references on this subject see e.g. Erdélyi 2006a; Pintér and Erdélyi 2011). In magnetic regions such as sunspots and pores, the amplitudes of the 5-min oscillations are suppressed, while high frequency oscillations, above the acoustic cut-off, have enhanced power. ...
Alfvén waves are considered to be viable transporters of the
non-thermal energy required to heat the Sun's quiescent atmosphere. An
abundance of recent observations, from state-of-the-art facilities, have
reported the existence of Alfvén waves in a range of
chromospheric and coronal structures. Here, we review the progress made
in disentangling the characteristics of transverse kink and torsional
linear magnetohydrodynamic (MHD) waves. We outline the simple, yet
powerful theory describing their basic properties in (non-)uniform
magnetic structures, which closely resemble the building blocks of the
real solar atmosphere.
... See a recent review by e.g. Pintér & Erdélyi (2011). ...
... These oscillations are seen in intensity, line of sight (LOS) velocity and LOS magnetic field. The source of the 5-minute oscillation is a result of forcing by the 5-minute (p-mode) global solar oscillation (Marsh & Walsh 2008), which forms the basis of helioseismology (Thompson 2006;Pintér & Erdélyi 2011). The 5-minute oscillations are typically seen in simple molecular and non-ionized metal lines, which form low in the umbral photosphere and are moderately suppressed not only in the penumbra, but also in the chromospheric atmosphere above the umbra (Bogdan & Judge 2006). ...
Aims. By focussing on the oscillations of the cross-sectional area and the total intensity of magnetic waveguides located in the lower solar atmosphere, we aim to detect and identify magnetohydrodynamic (MHD) sausage waves.
Methods. Capturing several high-resolution time series of magnetic waveguides and employing a wavelet analysis, in conjunction with empirical mode decomposition (EMD), makes the MHD wave analysis possible. For this paper, two sunspots and one pore (with a light bridge) were chosen as examples of MHD waveguides in the lower solar atmosphere.
Results. The waveguides display a range of periods from 4 to 65 min. These structures display in-phase behaviour between the area and intensity, presenting mounting evidence for sausage modes within these waveguides. The detected periods point towards standing oscillations.
Conclusions. The presence of fast and slow MHD sausage waves has been detected in three different magnetic waveguides in the solar photosphere. Furthermore, these oscillations are potentially standing harmonics supported in the waveguides that are sandwiched vertically between the temperature minimum in the lower solar atmosphere and the transition region. The relevance of standing harmonic oscillations is that their exploitation by means of solar magneto-seismology may allow insight into the sub-pixel resolution structure of photospheric MHD waveguides.
... However, the p-modes can be modified by the magnetic field (for reviews with references on this subject see e.g. Erdélyi 2006a; Pintér & Erdélyi 2011). In magnetic regions such as sunspots and pores, the amplitudes of the 5-min oscillations are suppressed, while high frequency oscillations, above the acoustic cut-off, have enhanced power. ...
Alfven waves are considered to be viable transporters of the non-thermal
energy required to heat the Sun's quiescent atmosphere. An abundance of recent
observations, from state-of-the-art facilities, have reported the existence of
Alfven waves in a range of chromospheric and coronal structures. Here, we
review the progress made in disentangling the characteristics of transverse
kink and torsional linear magnetohydrodynamic (MHD) waves. We outline the
simple, yet powerful theory describing their basic properties in (non-)uniform
magnetic structures, which closely resemble the building blocks of the real
solar atmosphere.
Mechanical waves are ubiquitous within solar plasmas, coming in the form of Magnetohydrodynamic
(MHD) waves. The Sun and its atmosphere are not a single
homogeneous medium and, rather, have complex structures stratified by gravity,
temperature gradients, complex magnetic field structures, subject to bulk flows and
even partial ionisation, with all of these affecting how oscillations propagate within
the plasma.
Magneto-acoustic gravity (MAG) waves have been investigated extensively within
solar physics, with three popular choices of analytical modelling for a Cartesian
coordinate system: a magnetic field parallel, perpendicular, or at an angle to the
gravitational field. Firstly we study the eigen-modes of bounded solar plasmas
embedded in a magnetic field perpendicular to the gravitational field and their energy
distribution in both single and two-layer models. We show that, indeed, modes can
still be split into fast and slow MAG with stratification decreasing the magnetic
energy with height and, thus, only waves with predominantly internal energy are
more evenly distributed in the atmosphere. A discontinuity in temperature between
layers also reflects waves but we show there is an inherent coupling between waves in
both layers. Secondly we investigate the effect of a bulk flow on MAG surface waves,
with the magnetic field parallel to the surface and a gravitational field perpendicular
to this. We find that waves along the penumbra, where Evershed flows are present
can change their direction of propagation and even the Kelvin-Helmholtz instability
can occur.
Waves within solar plasmas have generally been studied in fully ionised or
completely neutral media extensively and, as such, partial ionisation has not been
covered much until fairly recently, meaning there is great scope for exciting studies
in this sub-set of solar physics. Using single and multi-fluid methods we investigate
the stability of partially ionised plasma slabs with bulk flows. We find that in the
single fluid approximation that dissipative instabilities can occur for flow speeds
of the internal tube speed, with a neutrals providing stability. In the two-fluid
approximation our analysis confirms the existence of a mode that arises due to
the shear in flow in the neutral fluid and that, in a highly collisional plasma, has a
semi-resonant interaction between itself and the classical modes of incompressible or
compressible magnetic slabs.
Solar fundamental (f) acoustic mode oscillations are investigated analytically in a magnetohydrodynamic (MHD) model. The model consists of three layers in planar geometry, representing the solar interior, the magnetic atmosphere, and a transitional layer sandwiched between them. Since we focus on the fundamental mode here, we assume the plasma is incompressible. A horizontal, canopy-like, magnetic field is introduced to the atmosphere, in which degenerated slow MHD waves can exist. The global (f-mode) oscillations can couple to local atmospheric Alfvén waves, resulting, e.g., in a frequency shift of the oscillations. The dispersion relation of the global oscillation mode is derived, and is solved analytically for the thin-transitional layer approximation and for the weak-field approximation. Analytical formulae are also provided for the frequency shifts due to the presence of a thin transitional layer and a weak atmospheric magnetic field. The analytical results generally indicate that, compared to the fundamental value (ω=gk), the mode frequency is reduced by the presence of an atmosphere by a few per cent. A thin transitional layer reduces the eigen-frequencies further by about an additional hundred microhertz. Finally, a weak atmospheric magnetic field can slightly, by a few percent, increase the frequency of the eigen-mode. Stronger magnetic fields, however, can increase the f-mode frequency by even up to ten per cent, which cannot be seen in observed data. The presence of a magnetic atmosphere in the three-layer model also introduces non-permitted propagation windows in the frequency spectrum; here, f-mode oscillations cannot exist with certain values of the harmonic degree. The eigen-frequencies can be sensitive to the background physical parameters, such as an atmospheric density scale-height or the rate of the plasma density drop at the photosphere. Such information, if ever observed with high-resolution instrumentation and inverted, could help to gain further insight into solar magnetic structures by means of solar magneto-seismology, and could provide further insight into the role of magnetism in solar oscillations.
The linear theory of MHD resonant waves in inhomogeneous plasmas is reviewed. The review starts from discussing the properties
of driven resonant MHD waves. The dissipative solutions in Alfvén and slow dissipative layers are presented. The important
concept of connection formulae is introduced. Next, we proceed on to non-stationary resonant MHD waves. The relation between
quasi-modes of ideal MHD and eigenmodes of dissipative MHD are discussed. The solution describing the wave motion in non-stationary
dissipative layers is given. It is shown that the connection formulae remain valid for non-stationary resonant MHD waves.
The initial-value problem for resonant MHD waves is considered. The application of theory of resonant MHD waves to solar physics
is discussed.
KeywordsPlasma–Waves–MHD–Sun
