Changes in convective properties over the solar cycle: effect on p-mode damping rates

Measurements of both solar irradiance and p-mode oscillation frequencies indicate that the structure of the Sun changes with the solar cycle. Balmforth, Gough & Merryfield investigated the effect of symmetrical thermal disturbances on the solar structure and the resulting pulsation frequency changes. They concluded that thermal perturbations alone cannot account for the variations in both irradiance and p-mode frequencies, and that the presence of a magnetic field affecting acoustical propagation is the most likely explanation of the frequency change, in the manner suggested earlier by Gough & Thompson and by Goldreich et al. Numerical simulations of Boussinesq convection in a magnetic field have shown that at high Rayleigh number the magnetic field can modify the preferred horizontal length scale of the convective flow.
Here, we investigate the effect of changing the horizontal length scale of convective eddies on the linewidths of the acoustic resonant mode peaks observed in helioseismic power spectra. The turbulent fluxes in these model computations are obtained from a time-dependent, non-local generalization of the mixing-length formalism. The modelled variations are compared with p-mode linewidth changes revealed by the analysis of helioseismic data collected by the Birmingham Solar-Oscillations Network (BiSON); these low-degree (low- l ) observations cover the complete falling phase of solar activity cycle 22. The results are also discussed in the light of observations of solar-cycle variations of the horizontal size of granules and with results from 2D simulations by Steffen of convective granules.     

Distribution of magnetic flux on the solar surface and low-degree p-modes

The frequencies of solar p-modes are known to change over the solar cycle. There is also recent evidence that the relation between frequency shift of low-degree modes and magnetic flux or other activity indicators differs between the rising and falling phases of the solar cycle, leading to a hysteresis in such diagrams. We consider the influence of the changing large-scale surface distribution of the magnetic flux on low-degree ( l ≤3) p-mode frequencies. To that end, we use time-dependent models of the magnetic flux distribution and study the ensuing frequency shifts of modes with different order and degree as a function of time. The resulting curves are periodic functions (in simple cases just sine curves) shifted in time by different amounts for the different modes. We show how this may easily lead to hysteresis cycles comparable to those observed. Our models suggest that high-latitude fields are necessary to produce a significant difference in hysteresis between odd- and even-degree modes. Only magnetic field distributions within a small parameter range are consistent with the observations by Jiménez-Reyes et al. Observations of p-mode frequency shifts are therefore capable of providing an additional diagnostic of the magnetic field near the solar poles. The magnetic distribution that is consistent with the p-mode observations also appears reasonable compared with direct measurements of the magnetic field.     

Solar p modes of high degree l: coupling by differential rotation

Accurate measurements of solar p-mode frequencies and frequency splittings at high degree l require an adequate theoretical knowledge of the effects of mode coupling, induced by the variation with latitude of the angular velocity of the solar internal rotation. Earlier results for expansion coefficients of composite solutions (coupling coefficients) are due to Woodard. In this paper, the analysis is extended to allow for the dependence of the differential rotation on depth, and the result is expressed in terms of measurable quantities (the rotational splitting coefficients), which makes it convenient for diagnostic purposes. The analysis is based on the approach of quasi-degenerate perturbation theory, and is extended further to address possible effects of mode coupling in the observational line profiles. It is shown, using approximations applicable at high degree l , that the expected line profiles of composite modes in the observational power spectra are not distorted by mode coupling.     

On the measurement precision of solar p-mode eigenfrequencies

We make use of 3456 d of observations of the low-ℓ p-mode oscillations of the Sun in order to study the evolution over time of the measurement precision of the radial eigenfrequencies. These data were collected by the ground-based Birmingham Solar-Oscillations Network (BiSON) between 1991 January and 2000 June. When the power spectrum of the complete time series is fitted, the analysis yields frequency uncertainties that are close to those expected from the returned coherence times of the modes. The slightly elevated levels compared with the prediction appear to be consistent with a degradation of the signal-to-noise ratio in the spectrum that is the result of the influence of the window function of the observations (duty cycle 71 per cent). The fractional frequency precision reaches levels of a several parts in 10⁶ for many of the modes. The corresponding errors reported from observations made by the GOLF instrument on board the ESA/NASA SOHO satellite, when extrapolated to the length of the BiSON data set, are shown to be (on average) about ∼25 per cent smaller than their BiSON counterparts owing to the uninterrupted nature of the data from which they were derived.
An analysis of the BiSON data in contiguous segments of different lengths, T , demonstrates that the frequency uncertainties scale as T ^−1/2. This is to be expected in the regime where the coherence (life) times of the modes, τ _{n ℓ}, are smaller than the observing time T (the 'oversampled' regime). We show that mode detections are only now beginning to encroach on the 'undersampled' regime (where   T < τ _{n ℓ})  .     

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.     

Acoustic interferometry of the solar atmosphere: p-modes with frequencies near the 'acoustic cut-off'

High-frequency p-mode intensity data, obtained from the South Pole in 1987, 1988, 1990 and 1994, show a sharp variation in the phase-shift function and in the frequency spacings near 5.5 mHz. Using a simple theoretical model, we demonstrate that this behaviour is caused by an acoustic resonance in the atmosphere between the excitation source and the upper reflection level. We discuss the diagnostic properties of this resonance, which is sensitive to the acoustic reflectivity of the solar atmosphere and to the location and parity of the excitation source. When applied to the solar data, our model indicates that the average acoustic reflectivity increases with increasing solar activity. The model also shows that the acoustic source has composite parity and is located within one pressure scaleheight of the base of the photosphere.     

Seismology of stellar envelopes: probing the outer layers of a star through the scattering of acoustic waves

The outer layers of Sun-like stars are regions of rapid spatial variation which modulate the p-mode frequencies by partially reflecting the constituent acoustic waves. With the accuracy that has been achieved by current solar observations, and that is expected from imminent stellar observations, this modulation can be observed from the spectra of the low-degree modes. We present a new and simple theoretical calculation to determine the leading terms in an asymptotic expansion of the outer phase of these modes, which is determined by the structure of the surface layers of the star. Our procedure is to compare the stellar envelope with a plane-parallel polytropic envelope, which we regard as a smooth reference background state. Then we can isolate a seismic signature of the acoustic phase and relate it to the stratification of the outer layers of the convection zone. One can thereby constrain theories of convection that are used to construct the convection zones of the Sun and Sun-like stars. The accuracy of the diagnostic is tested in the solar case by comparing the predicted outer phase with an exact numerical calculation.     

Effects of a gap-filling method on p-mode parameters

The quality of helioseismological ground-based data strongly depends on the presence of a gap in the observational window. In order to address that problem in the case of full-disc low-degree p-mode velocity measurements, Fossat et al. proposed a gap-filling method called 'Repetitive music'. The autocorrelation function of the velocity signal shows a correlation of more than 70 per cent at about 4 h because of the quasi-periodicity of p-mode peaks in the Fourier spectrum. The method then consists of filling gaps of the velocity signal with data, when they exist, located 4 h before or after.
By using Monte Carlo simulations, we assess the effects of the gap-filling method on p-mode parameters and their errors. A way to remove the modulation in the power spectrum resulting from the gap-filling method is proposed; its effects on p-mode frequencies, linewidths, amplitudes and asymmetries are discussed as a function of frequency and signal-to-noise ratio.     

Rigid rotation of the solar core? On the reliable extraction of low-ℓ rotational p-mode splittings from full-disc observations of the Sun

We present low-ℓ rotational p-mode splittings from the analysis of 8 yr of observations made by the Birmingham Solar-Oscillations Network (BiSON) of the full solar disc. These data are presented in the light of a thorough investigation of the fitting techniques used to extract them. Particular attention is paid to both the origin and magnitude of bias present in these estimates. An extensive Monte Carlo strategy has been adopted to facilitate this study – in all, several thousand complete, artificial proxies of the 96-month data set have been generated to test the analysis of real 'full-disc' data. These simulations allow for an assessment of any complications in the analysis which might arise from variations in the properties of the p modes over the 11-yr solar activity cycle.
The use of such an extended data set affords greater precision in the splittings, and by implication the rotation rate inferred from these data, and reduces bias inherent in the analysis, thereby giving a more accurate determination of the rotation. The grand, weighted sidereal average of the BiSON set is     , a value consistent with that expected were the deep radiative interior     to rotate at the same frequency, and in the same 'rigid' manner, as the more precisely and accurately studied outer part of the radiative zone.     

Probing sunspot magnetic fields with p-mode absorption and phase shift data

Long-standing observations of incoming and outgoing f- and p-modes in annuli around sunspots reveal that the spots partially absorb and substantially shift the phase of waves incident upon them. The commonly favoured absorption mechanism is partial conversion to slow magneto-acoustic waves that disappear into the solar interior channelled by the magnetic field of the sunspot. However, up until now, only f-mode absorption could be accounted for quantitatively by this means. Based on vertical magnetic field models, the absorption of p-modes was insufficient. In this paper, we use the new calculations of Crouch & Cally for inclined fields, and a simplified model of the interaction between spot interior and exterior. We find excellent agreement with phase shift data assuming field angles from the vertical in excess of 30° and Alfvén/acoustic equipartition depths of around 600–800 km. The absorption of f-modes produced by such models is considerably larger than is observed, but consistent with numerical simulations. On the other hand, p-mode absorption is generally consistent with observed values, up to some moderate frequency dependent on radial order. Thereafter, it is too large, assuming absorbing regions comparable in size to the inferred phase-shifting region. The excess absorption produced by the models is in stark contrast with previous calculations based on a vertical magnetic field, and is probably due to finite mode lifetimes and excess emission in acoustic glories. The excellent agreement of phase shift predictions with observational data allows some degree of probing of subsurface field strengths, and opens up the possibility of more accurate inversions using improved models. Most importantly, though, we have confirmed that slow mode conversion is a viable, and indeed the likely, cause of the observed absorption and phase shifts.     

Use of genetic algorithms to determine low-l rotational p-mode splittings at high frequencies

We present a comparative study of genetic and standard fitting routines applied to the task of extracting reliable estimates of the rotational splitting of full-disc, low-angular-degree (low- l ) solar p-mode data at high frequencies. 100 artificial proxies of a 10-yr data base of observations made by the Birmingham Solar-Oscillations Network (BiSON) were used to test the two approaches. All sets were analysed over the frequency range from 3000 to 4000 μHz.
Previous work, based on non-linear 'hill-climbing' fitting techniques, has demonstrated the unfortunate tendency for full-disc estimates of the splitting to overestimate the true, underlying values at high frequencies. Here, we show that the resulting bias is less severe when a genetic-fitting approach is adopted. This is largely the result of the number of erroneous 'null-valued' estimates of the splitting being considerably reduced: these estimates are, in effect, re-introduced into the expected normal distribution of fitted splittings. We also illustrate the diverse control one has when using a genetic algorithm as a fitting routine; this diversity is shown to allow further refinement in the estimate of the rotational splitting.
Finally, we address the issue of the reliability of the formal splitting uncertainties returned by the mode fitting, and find that complications arising from the strong anticorrelation between the splittings and their error estimates are not alleviated by the use of the genetic technique.     

On the choice of parameters in solar-structure inversion

The observed solar p-mode frequencies provide a powerful diagnostic of the internal structure of the Sun and permit us to test in considerable detail the physics used in the theory of stellar structure. Among the most commonly used techniques for inverting such helioseismic data are two implementations of the optimally localized averages (OLA) method, namely the subtractive optimally localized averages (SOLA) and multiplicative optimally localized averages (MOLA). Both are controlled by a number of parameters, the proper choice of which is very important for a reliable inference of the solar internal structure. Here we make a detailed analysis of the influence of each parameter on the solution and indicate how to arrive at an optimal set of parameters for a given data set.     

Magneto‐seismology of the solar atmosphere

Magnetohydrodynamic (MHD) waves in solar coronal loops, which were previously only predicted by theory have actually been detected with space‐borne instruments. These observations have given an important and novel tool to measure fundamental parameters in the magnetically embedded solar corona. This paper will illustrate how information about the magnetic and density structure along coronal loops can be determined by measuring the frequency or amplitude profiles of standing fast kink mode oscillations. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Variations of the low‐degree solar p‐mode frequencies for 10 years of GOLF observations

We experiment with a method of measuring the frequency of solar p modes, intended to extend the passband for the variations of the frequency spectrum as high as possible. So far this passband is limited to a fraction of μ Hz for the classical analysis based on numerical fits of a theoretical line profile to a power spectrum averaged over periods lasting at least several weeks. This limit for the present analysis can be shifted to the mHz range, corresponding to some of the “5 min” oscillations, but in this range we use a lower resolution which allows us to separate odd and even p modes. We show an example of the results for long term variations and apply this analysis to search for a modulation of the p‐mode frequency spectrum by asymptotic series of solar g modes. A faint signal is found in the analysis of 10 years of GOLF data. This very preliminary result possibly indicates the detection of a small number of g modes of degree l = 1. A tentative determination of an observational value of the parameter P₀ follows. P₀ is the scaling factor of the asymptotic series of g modes and is a key data for solar core physics. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On model predictions of the power spectral density of radial solar p modes

We investigate the frequency dependence of the power spectral density of low-degree solar p modes by comparing measurements with the results of a stochastic-excitation model. In the past it was common practice to use the total power in such investigations. Using the maximum of the power spectral density instead provides a direct comparison with the measured mode heights in the observed power spectrum. This method permits a more careful calibration of the adjustable parameters in the excitation model, a model which we present here, for the first time, in a format that precisely and unambiguously relates the amplitudes of the modes of oscillation to the Reynolds stress in the equilibrium model. We find that errors in the theory of the linear mode damping rates, particularly at low frequency, have a dramatic impact on the predictions of the mode heights in the spectral density, whereas parameter changes in the stochastic excitation model, within a plausible domain of parameter space, have a comparatively small effect.     

p‐mode power variation with solar atmosphere as observed in the Na D1 and K spectral lines

In this work we investigate p‐mode power variation with solar atmosphere. To this aim, we use THÉMIS observations of the Na D1 (λ 5896 Å) and K (λ 7699 Å) spectral lines. While the formation heights of the K spectral line are essentially located in the photospheric layer, the formation heights of the Na D1 line span a much wider region: from photosphere up to chromosphere. Hence, we had the opportunity to infer p‐mode power variation up to the chromospheric layer. By analyzing power spectra obtained by temporal series at different points of the Na D1 and K spectral lines, we confirm and quantify the increase in p‐mode power towards higher atmospheric layers. Furthermore, the large span in formation heights of the Na D1 line induces a larger enhancement of p‐mode power with solar atmosphere compared to the K spectral line. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Anisotropy and Dissipation of Turbulence and Their Effects on Solar Models

1 INTRODUCTIONThe maing-length theory (MLT) is the most commonly used approach to calculate convective energy transport in stars and other astrophysical situations. Based on the original idea ofPrandtl (1952) that turbulent parcels trallsfer heat in a similar way as molecules of gas do inthermal conduction, the MLT assumes that convection cells, drived by buoyancy, move thlougha ~ng length 1 and release the heat they carry when they merge with their environment. Themost widely adopted f…