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.     

Current convection in solar active regions

Current carrying magnetic fields which penetrate sunspots can be unstable to current convective modes caused by the large gradient of electrical conductivity. The linear growth rates and wavelengths of the unstable modes are found. The unstable modes produce fine-scale vortices perpendicular to the magnetic field, which overshoot well into the solar corona. The modes provide a turbulent vorticity source at the photospheric footpoints of the field. This can cause braiding and reconnection of the coronal magnetic field. The modes twist the coronal magnetic field into loops with a typical radius of 200 km, consistent with recent X-ray observations.     

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)     

Probing solar convection

In the solar convection zone, acoustic waves are scattered by turbulent sound speed fluctuations. In this paper the scattering of waves by convective cells is treated using Rytov's technique. Particular care is taken to include diffraction effects, which are important, especially for high-degree modes that are confined to the surface layers of the Sun. The scattering leads to damping of the waves and causes a phase shift. Damping manifests itself in the width of the spectral peak of p-mode eigenfrequencies. The contribution of scattering to the linewidths is estimated and the sensitivity of the results to the assumed spectrum of the turbulence is studied. Finally, the theoretical predictions are compared with recently measured linewidths of high-degree modes.     

A spherical harmonics filter for solar oscillations research

A scheme, based on the expansion of solar oscillations into spherical harmonics, for the identification of sectorial modes of intermediate degree in the interval 3 < l < 20 is presented. In this range, the frequencies of modes with similar quantum numbers can be very close together, so that a careful spectral analysis of their spatial pattern is needed to effectively separate these modes. The filtering scheme proposed is intended to operate on quantized images of the Sun and reaches satisfactory resolving power by a two-step procedure, namely a straightforward filtering followed by the resolution of a system of linear equations. The results obtained are also shown to be independent on the ecliptic longitude of the Earth.     

Rotation of the solar core from BiSON and LOWL frequency observations

Determination of the rotation of the solar core requires very accurate data on splittings for the low-degree modes which penetrate to the core, as well as for modes of higher degree to suppress the contributions from the rest of the Sun to the splittings of the low-degree modes. Here we combine low-degree data based on 32 months of observations with the BiSON network and data from the LOWL instrument. The data are analysed with a technique that specifically aims at obtaining an inference of rotation that is localized to the core. Our analysis provides what we believe is the most stringent constraint to date on the rotation of the deep solar interior.     

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.     

Trapped gravity waves and the five-minute oscillations of the solar atmosphere

The various modes of hydrodynamic waves are considered for a model of the solar atmosphere which is based on the Bilderberg model and includes the effects of ionization. The atmosphere forms a potential well for internal gravity waves, since the stability is low at the base (near the convection) and low again in the region of partial ionization in the chromosphere. Calculations show that there are two resonant (trapped) modes of internal gravity waves for horizontal wavelengths based on the scale of the granulation. The properties of these modes are in close agreement with the two modes of oscillation observed by Frazier (1968). Trapped acoustic modes are found to have periods too short to account for the observations.Presently Visiting Fellow, Joint Institute for Laboratory Astrophysics, University of Colorado, Boulder, Colo.     

About the rotation of the solar radiative interior

In the modern era of helioseismology we have a wealth of high-quality data available, e.g., more than 6 years of data collected by the various instruments on board the SOHO mission, and an even more extensive ground-based set of observations covering a full solar cycle. Thanks to this effort a detailed picture of the internal rotation of the Sun has been constructed. In this paper we present some of the actions that should be done to improve our knowledge of the inner rotation profile discussed during the workshop organized at Saclay on June 2003 on this topic. In particular we will concentrate on the extraction of the rotational frequency splittings of low- and medium-degree modes and their influence on the rotation of deeper layers. Furthermore, for the first time a full set of individual |m|-component rotational splittings is computed for modes ℓ≤4 and 1<ν<2 mHz, opening new studies on the latitudinal dependence of the rotation rate in the radiative interior. It will also be shown that these splittings have the footprints of the differential rotation of the convective zone which can be extremely useful to study the differential rotation of other stars where only these low-degree modes will be available.     

Observations of low-degree solar oscillations with few detector elements

The detection of low-degree solar oscillation modes with a specific low-resolution detector configuration is investigated. The detector is part of an instrument (the Luminosity Oscillations Imager) in the VIRGO package, to be flown on SOHO. Various problems such as p- and g-mode sensitivity, B and roll angle effects, modes isolation, cross-talk and guiding effects are treated for a given detector configuration. The computed sensitivity will enable the instrument to detect any type of modes for l < 6.B and roll angle effects can be compensated by using adequate filters for mode isolation. Guiding effects are small for p-modes. Also some other complex high-degree mode effects are treated.     

Excitation of high-m tearing modes at the solar flare site

It is shown that high-m drift tearing modes can be excited under the conditions prevalent at the solar flare sites. Since the growth rate of the high-m tearing modes is larger than that for low-m macroscopic tearing modes and smaller than that of microscopic ion-acoustic instability, these modes warrant accommodation in the scheme of instabilities possibly operating in the hybrid model of solar flares suggested by Spicer.     

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.     

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)     

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.     

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})  .     

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.     

Non-divergent oscillations in the solar atmosphere

Non-divergent oscillations having the form of deep water waves are shown to form normal modes or free oscillations of the solar atmosphere under two approximations: the chromosphere-coronal interface behaves like a free surface, and the density scale height is sufficiently large in the convective zone. These modes show the temporal and spatial characteristics of the 300 second chromospheric oscillations.     

The potential of solar high-degree modes for structure inversion

It is likely that precise and reliable frequencies of high-degree modes will soon be available from the SOI/MDI experiment. Here we examine the ability of such modes (with l>300) to resolve the solar structure in the near-surface region. In particular, we investigate inversions to determine the adiabatic exponent ₁ as a test of the solar equation of state, as well as the potential of such data to constrain the solar envelope helium abundance.     

Prediction of the height of solar cycle 23

Presuming a bimodal behaviour of even-odd solar cycle pairs (i.e., four modes designated asA, B, C, andD), we predict the amplitude of solar cycle 23. The bimodal properties include the dependence of maximum relative sunspot number (RM) on cycle rise time (TR) separately for odd-following and even cycles (both in two split modes), and the dependencies of odd-following on even cycles separately for cycle rise times and maximum relative sunspot numbers (each also split into two mode pairs). The procedure was first to identify the proper mode for cycle 22 (modeA), which then explicitly defines the mode for cycle 23 (modeC). The presumed mode-inherent relations were then used to estimate the rise time for cycle 23 (3.7 0.5 yr) and its maximum amplitude (195.1 17.1). A second estimate of maximum amplitude, based directly on a presumed mode-inherent relation between maximum amplitudes for even and odd cycle pairs, yields a somewhat lower value (181.3 44.3). Thus, the results of this analysis supports previous findings that cycle 23 may be one of the largest amplitude cycles ever observed.