Influence of a chromospheric magnetic field on solar acoustic modes

This paper studies the effect of a magnetic atmosphere on the global solar acoustic oscillations in a simple Cartesian model. First, the influence of the ratio of the coronal and the photospheric temperature τ and the strength of the magnetic field at the base of the corona B_c on the oscillation modes is studied for a convection zone-corona model with a true discontinuity. The ratio τ seems to be an important parameter. Subsequently, the discontinuity is replaced by an intermediate chromospheric layer of thickness L and the effect of the thickness on the frequencies of the acoustic waves is studied. In addition, nonuniformity in the magnetic field, plasma density and temperature in the transition layer gives rise to continuous Alfvén and slow spectra. Modes with characteristic frequencies lying within the range of the continuum may resonantly couple to Alfvén and/or slow waves.     

Overstability of acoustic modes and the solar five-minute oscillations

The stability of linear convective and acoustic modes in solar envelope models is investigated by incorporating the thermal and mechanical effects of turbulence through the eddy transport coefficients. With a reasonable value of the turbulent Prandtl number it is possible to obtain the scales of motion corresponding to granulation, supergranulation and the five-minute oscillations. Several of the acoustic modes trapped in the solar convection zone are found to be overstable and the most unstable modes, spread over a region centred predominantly around a period of 300 s with a wide range of horizontal length scales, are in reasonable accord with the observed power-spectrum of the five-minute oscillations. It is demonstrated that these oscillations are driven by a simultaneous action of the -mechanism and the radiative and turbulent conduction mechanisms operating in the strongly superadiabatic region in the hydrogen ionization zone, the turbulent transport being the dominant process in overstabilizing the acoustic modes.     

The visibility of low‐frequency solar acoustic modes

We make predictions of the detectability of low‐frequency p modes. Estimates of the powers and damping times of these low‐frequency modes are found by extrapolating the observed powers and widths of higher‐frequency modes with large observed signal‐to‐noise ratios. The extrapolations predict that the low‐frequency modes will have small signal‐to‐noise ratios and narrow widths in a frequency‐power spectrum. Monte Carlo simulations were then performed where timeseries containing mode signals and normally distributed Gaussian noise were produced. The mode signals were simulated to have the powers and damping times predicted by the extrapolations. Various statistical tests were then performed on the frequency‐amplitude spectra formed from these timeseries to investigate the fraction of spectra in which the modes could be detected. The results of these simulations were then compared to the number of p‐modes candidates observed in real Sun‐as‐a‐star data at low frequencies. The fraction of simulated spectra in which modes were detected decreases rapidly as the frequency of modes decreases and so the fraction of simulations in which the low‐frequency modes were detected was very small. However, increasing the signal‐to‐noise (S/N) ratio of the low‐frequency modes by a factor of 2 above the extrapolated values led to significantly more detections. Therefore efforts should continue to further improve the quality of solar data that is currently available. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

An attempt to identify low l - low n solar acoustic modes

The low l solar acoustic spectrum has been measured with great accuracy (v/v 10^–4), for intermediate radial order modes, 11 n 34 (Jiménez et al., 1986; Grec, Fossat, and Pomerantz, 1983; Pallé et al., 1986). The measurement of the frequencies of modes of lower n, up to the fundamental one, are very important as they depart from asymptotic behaviour and, therefore, put more severe constraints on solar models. However, their amplitudes are very low (under 2 cm s^–1) and when compared to the solar velocity background noise (Jiménez et al., 1986), a S/N 1 is obtained. Taking advantage of the fact that lifetimes seem to be higher at lower frequencies (lower n values) (Jefferies et al., 1988; Elsworth et al., 1990), very long Doppler velocity measurements, obtained at Teide Observatory, have been used to increase S/N, therefore, providing the possibility to detect such modes. The frequencies observed are compared to those predicted by a solar model (Christensen-Dalsgaard, Däppen, and Lebreton, 1988), using the best equation of state yet computed (Mihalas, Däppen, and Hummer, 1988).     

Phase differences between irradiance and velocity low degree solar acoustic modes revisited

Since 1984, simultaneous observations of irradiance and velocity solar acoustic modes, have been carried out by several authors in order to measure the phase difference between irradiance and velocity modes. Following the earliest observations with stratospheric balloon (Frolich and van Der Raay, 1984), a two ground-based stations (Tenerife and Baja California) were established (Jimenez et al, 1990) obtaining coherence results in the frequency range from 2.5 mHz to 4.3 mHz. These phase differences between irradiance and velocity solar acoustic modes are interpreted in terms of the non-adiabatic behaviour of the solar atmosphere. In 1988 the IPHIR (Frolich et al, 1988) instrument flown on the PHOBOS-2 mission to Mars and measured the solar irradiance during 150 consecutive days. The best velocity observations obtained in Tenerife for this period were compared with IPHIR data to compute the phase differences (Schrijver et al, 1991). The final conclusion is that good agreement is attained between space quadsi-space and ground observations which yield a phase diffenrece of about -125 degrees in the frequency range 2.5 mHz to 4.2 mHz, with a slight increase suggested by the data running up to 4.6 mHz.     

On the variations in the parameters of high-degree acoustic modes with solar cycle

We analyze the pattern of behavior of p-mode wave packets with solar cycle using TON one-day helioseismic data with a high spatial resolution. The time—distance method is used to perform this task. We make an attempt to determine the variations in the travel time of acoustic waves at maximum and minimum solar activity; at maximum activity, this time decreases by 2 s compared to that at minimum activity to a depth of 0.8R_⊙. In addition, the correlation amplitudes of acoustic wave packets from minimum to maximum solar activity were found to decrease by 10–20% for all angular distances.     

Star-formation modes in the solar neighborhood

We perform a study into the spatial and kinematical distribution of young open clusters in the solar neighborhood, distinguishing between Gould Belt and local Galactic disk members. We use a previous estimate of the structural parameters of both systems obtained from a sample of O to B6 stars from Hipparcos. The two star-forming regions that dominate and give the Gould Belt its characteristic inclined shape show a striking difference in their content of star clusters: while Ori OB1 is richly populated by open clusters, not a single one can be found within the boundaries of Sco OB2. This is mirrored in velocity space, translating again into an abundance of clusters in the region of the kinematic space populated by the members of Ori OB1, and a marginal number of them associated with Sco OB2. We interpret all these differences by characterizing the Orion region as a cluster complex typically surrounded by a stellar halo, and the Sco-Cen region as an OB association in the outskirts of the complex. The different contents of star clusters, the different heights above the Galactic plane and the different residual velocities of Ori OB1 and Sco OB2 can be explained in terms of their relative position with respect to the density maximum of the Local Arm in the solar neighborhood. The origin of this feature could have been the interaction of a density wave with the local interstellar medium close to the Galactic co-rotation radius.     

Modelling solar atmospheric gravity oscillation modes

Solar oscillations are investigated in a one‐dimensional hydrodynamic plane‐parallel model with an atmosphere. Besides the acoustic pressure (p) modes, the fundamental (f) and Lamb mode, another set of eigenmodes, a group of atmospheric gravity (g) modes, is found in the low‐frequency region of the spectrum. Their frequencies and spatial behaviour are studied. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Resonant coupling between solar gravity modes

It is shown that in consequence of the parametric resonance, g modes of low spherical harmonic degree l are strongly coupled to the modes of high degree. The coupling limits the growth of low l modes to very small amplitudes. For g ₁, l = 1 mode, the final amplitude of the radial velocity is of the order of 10 cm s^-1. A mixing of solar core as a result of a finite-amplitude development of linear instability of this mode is thus highly unlikely.     

Influence of a uniform coronal magnetic field on solar p modes: coupling to slow resonant MHD waves

The influence of a constant coronal magnetic field on solar global oscillations is investigated for a simple planar equilibrium model. The model consists of an atmosphere with a constant horizontal magnetic field and a constant sound speed, on top of an adiabatic interior having a linear temperature profile. The focus is on the possible resonant coupling of global solar oscillation modes to local slow continuum modes of the atmosphere and the consequent damping of the global oscillations. In order to avoid Alfvén resonances, the analysis is restricted to propagation parallel to the coronal magnetic field. Parallel propagating oscillation modes in this equilibrium model have already been studied by Evans and Roberts (1990). However, they avoided the resonant coupling to slow continuum modes by a special choice of the temperature profile. The physical process of resonant absorption of the acoustic modes with frequency in the cusp continuum is mathematically completely described by the ideal MHD differential equations which for this particular equilibrium model reduce to the hypergeometric differential equation. The resonant layer is correctly dealt with in ideal MHD by a proper treatment of the logarithmical branch cut of the hypergeometric function. The result of the resonant coupling with cusp waves is twofold. The eigenfrequencies become complex and the real part of the frequency is shifted. The shift of the real part of the frequency is not negligible and within the limit of observational accuracy. This indicates that resonant interactions should definitely be taken into account when calculating the frequencies of the global solar oscillations.     

Global modes constituting the solar magnetic cycle

The spherical-harmonic-Fourier analysis of the Sun's magnetic field inferred from the Greenwich sunspot data is refined and extended to include the full length (1874–1976) of the data on the magnetic tape provided by H. Balthasar. Perspective plots and grey level diagrams of the SHF power spectra for the odd and the even degree axisymmetric modes are presented. Comparing these with spectra obtained from two simulated data sets with random redistribution within the wings in the butterfly diagrams, we conclude that there is no clear evidence for the existence of any relation between the harmonic degree and the temporal frequency of the power concentrations of the inferred field. Apart from the power ridge in the narrow frequency band at 1/21.4 y ^–1, and low ridges at odd multiples of this frequency, there are no other spectral features. This strongly suggests that the solar magnetic cycle consists of some global oscillations of the Sun forced at a frequency 1/21.4 y ^–1 and, perhaps, weak resonances at its odd harmonics. The band width of the forcing frequency seems to be much less than 1/107 y ^–1. In case the global oscillations are torsional MHD, the significance of their parity and power peak is pointed out.     

Global modes constituting the solar magnetic cycle

The sunspot occurrence probability defined in Paper I is used to determine the Legendre-Fourier (LF) terms in the rate of emergence of toroidal magnetic flux,Q(, t), above the photosphere per unit latitude interval, per unit time. Assuming that the magnetic flux tubes whose emergence yields solar activity are produced by interference of global MHD waves in the Sun, we determine how the amplitudes and phases of the LF terms in the toroidal magnetic fieldB , representing the waves, will be related to those of the LF terms inQ(, t). The set of LF terms in Q that represents the set of waves whose interference produces most of the observed sunspot activity is {l = 1, 3, , 13;v =nv _*,n = 1, 3, 5}, wherev _* = 1/21.4 yr^–1. However, among the shapes of sunspot cycles modeled using various sets of the computed LF terms the best agreement with the observed shape, for each cycle, is given by the set {l = 3 orl = 3, 5; andn = 1, 3 orn = 1, 3, 5}. The sets of terms: {l = 1, 3, 5, 7;n = 1}, {l = 1, 3, 5, 7;n = 3}, {l = 9, 11, 13, 15;n = 1} and {l = 9, 11, 13, 15;n = 3} seem to represent four modes of global MHD oscillation. Correlations between the amplitudes (and phases) of LF terms in different modes suggest possible existence of cascade of energy from constituent MHD waves of lowerl andn to those of higherl andn. The spectrum of the MHD waves trapped in the Sun may be maintained by the combined effect of this energy cascade and the loss of energy in the form of the emerging flux tubes. The primary energy input into the spectrum may be occurring in the mode {l = 1, 3, 5, 7;n = 1). As expected from the above phenomenological model, the size of a sunspot cycle and its excess over the previous cycle are well correlated (e.g., 90%) to the phase-changes of the two most dominant oscillation modes during the previous one or two cycles. These correlations may provide a physical basis to forecast the cycle sizes.     

Global modes constituting the solar magnetic cycle

We show that the axisymmetric odd degree SHF modes of 21.4-yr periodicity and degrees l 29 in the solar magnetic field (as inferred from sunspot data during 1874–1976), are at least approximately stationary. Among the sine and cosine components of these SHF modes we find four groups, each defining the geometry of a coherent global oscillation characterized by a distinct power hump and its own level of variation. The first two of these geometrical eigenmodes (viz., B ₁ and B ₂), define the large-scale structure of the butterfly diagrams. Remaining SHF modes define the orderliness of the field distribution even within the wings of the butterflies down to scales l 29. These include the geometrical eigenmodes B ₃ and B ₄, which are not present in simulated data sets in which the latitudes of the sunspot groups are randomly redistributed within the wings of the butterflies.Superposition of B ₁, B ₂, B ₃, and B ₄ is necessary and sufficient to reproduce important observed properties of the latitude-time distribution of the real field, not only in the sunspot zone, but also in the middle (35°–75°) and the high (75°) latitudes, with appropriate relative orders of magnitude and phases. Thus, B ₁, B ₂, B ₃, and B ₄ seem to represent really existing global oscillations in the Sun's internal magnetic field. The geometrical form of B ₁ may also be the form of the forcing oscillation.     

The quest for the solar g modes

Solar gravity modes (or g modes)—oscillations of the solar interior on which buoyancy acts as the restoring force—have the potential to provide unprecedented inference on the structure and dynamics of the solar core, inference that is not possible with the well-observed acoustic modes (or p modes). The relative high amplitude of the g-mode eigenfunctions in the core and the evanesence of the modes in the convection zone make the modes particularly sensitive to the physical and dynamical conditions in the core. Owing to the existence of the convection zone, the g modes have very low amplitudes at photospheric levels, which makes the modes extremely hard to detect. In this article, we review the current state of play regarding attempts to detect g modes. We review the theory of g modes, including theoretical estimation of the g-mode frequencies, amplitudes and damping rates. Then we go on to discuss the techniques that have been used to try to detect g modes. We review results in the literature, and finish by looking to the future, and the potential advances that can be made—from both data and data-analysis perspectives—to give unambiguous detections of individual g modes. The review ends by concluding that, at the time of writing, there is indeed a consensus amongst the authors that there is currently no undisputed detection of solar g modes.     

Peak asymmetry of solar p modes: a framework to explain the effects of the correlated noise from the acoustic source

It is well known that in the power spectrum solar p modes have asymmetric profiles, which depart from a Lorentzian shape. We present a framework to explain the contribution of correlated background noise, from the acoustic source, to this asymmetry. An important prediction is that observed peak asymmetry may differ depending on the way the p-mode observations are made, and on how the data are prepared. Furthermore, if valid, the proposed framework may provide the basis for separating the contribution of the correlated noise from that of the source location and properties.     

Variation of the low-degree solar acoustic mode parameters over the solar cycle

VIRGO/SPM is a helioseismic sunphotometer on board SOHO that observes the disk-integrated sunlight irradiance at three different colors (red, green, and blue). The data obtained for SPM since the beginning of the SOHO mission, April 1996, to March 2001 have been used to study the differences of the p-mode parameters during the solar activity cycle. These time series have been divided in sub-series of 100 days, transformed to power spectra and averaged in sets of three to yield a total number of six averaged power spectra (around one per year). A new way of analyzing the power spectrum has been applied to the six power spectra of each color; it consists of fitting the whole p-mode spectrum at once with a unique background. The results for the frequencies, line widths, power, mode energy, energy rate fed in the mode and splittings along the activity cycle are found, compared and discussed.     

Magnetic perturbations to the acoustic modes of roAp stars

We study the effect of a large-scale surface magnetic field on the non-radial acoustic modes of roAp stars. Special attention is given to the use of a variational principle which is used for determining the shifts in the frequencies with relative ease, enabling us to avoid having to calculate the perturbed eigenfunctions. With knowledge of the frequency shifts we then estimate the eigenfunctions in a simpler, albeit approximate way. The results indicate frequency shifts of the order of few μHz, which depend on the order, degree and azimuthal order of the mode. The loss of energy through Alfvén waves is also estimated from the imaginary parts of the frequency shifts. The results indicate that the loss is particularly high near specific frequencies. This might indicate the presence of a selection effect, which could make some modes more likely to be excited than others. However, our results do not explain why the modes observed appear always to be aligned with the axis of the magnetic field. Finally, the estimated perturbed eigenfunctions contain strong components of spherical harmonics that differ from those of the original unperturbed modes.     

Investigating the excitation of acoustic modes using homomorphic deconvolution

Properties of the convection zone are encrypted in the behavior of acoustic modes (p modes) and also in the excitation mechanism. The goal of this study is to reconstruct the pattern of the excitation (e.g., the rate and strength of the excitation) from an observed signal. The method, homomorphic deconvolution, has been tested on several sets of artificial signals. The preliminary results presented here are promising. Further rigorous tests and the eventual application to a real signal will be carried out in the near future.     

Rigidly rotating modes of the solar magnetic field

Discrete rigidly rotating components (modes) of the large-scale solar magnetic field have been investigated. We have used a specially calculated basic set of functions to resolve the observed magnetic field into discrete components. This adaptive set of functions, as well as the expansion coefficients, have been found by processing a series of digitized synoptic maps of the background magnetic field over a 20-year period. As a result, dependences have been obtained which describe the spatial structure and the temporal evolution of the 27-day and 28-day rigidly rotating modes of the Sun's magnetic field.The spatial structure of the modes has been compared with simulations based on the known flux-transport equation. In the simulations, the rigidly rotating modes were regarded as stationary states of the magnetic field whose rigid rotation and stability were maintained by a balance between the emergence of magnetic flux from stationary sources located at low latitudes and the horizontal transport of flux by turbulent diffusion and poleward directed meridional flow. Under these assumptions, the structure of the modes is determined solely by the horizontal velocity field of the plasma, except for the low-latitude zone where sources of magnetic flux concentrate. We have found a detailed agreement between the simulations and the results of the data analysis, provided that the amplitude of the meridional flow velocity and the diffusion constant are equal to 9.5 m s^–1 and 600 km² s^–1, respectively.The analysis of the expansion coefficients has shown that the rigidly rotating modes undergo rapid step-like variations which occur quasi-periodically with a period of about two years. These variations are caused by separate surges of magnetic flux in the photosphere, so that each new surge gives rise to a rapid replacement of old large-scale magnetic structures by newly arisen ones.     

Two-telescope-based solar seeing profile measurement simulation

Measurements of the daytime seeing profile of the atmospheric turbulence are crucial for evaluating a solar astronomical site so that research on the profile of...