The investigation of the models of the contemporary Sun with a mixed core has shown that the amplitude of some gravity modes of oscillations of the star can be mainly concentrated in the central region. This phenomenon takes place if the node of the amplitude of radial displacement coincides with the boundary of the mixed core. In this case the core can be regarded as a driving generator of the oscillations, determining their period and phase. It is suggested as the explanation of the observational properties of the 160-min oscillation.

Normal mode spectra and neutrino counting rates are calculated for a set of chemically-inhomogeneous solar models. Each model has a core with a high concentration of heavy elements; high opacity makes the core convective. The structure of the envelope is that of the standard model. It is shown that (1) the spectrum of g modes becomes less densely separated than that of the standard model, which simplifies the problem of interpreting 160-min oscillations; (2) low neutrino counting rates may be achieved for a low initial helium concentration in the core; (3) the models do not contradict the frequency spacing of global 5-min oscillations.

Evolutionary models of the present Sun with standard and artifically low opacity of stellar matter are obtained and adiabatic nonradial oscillations of the models are computed. The opacity x in nonstandard model (which is the first in a series of future models with low χ) was taken half as much as the Cox-Stewart opacity. The central temperature in such model is approximately 10% below the ‘standard’ values, which can explain the neutrino experiment results. Unlike solar models with very low heavy element abundance, the low χ model has approximately standard mass concentration and distribution of the matter in the outer layers – specifically, the standard characteristics of the convection zone. Hence, the spectrum of adiabatic oscillations is similar to that of the standard models and has the same capabilities for the explanation of the observed pulsations.

We have calculated eigenfrequencies of radial and nonradial p-mode oscillations with low harmonic index l (l = 0, 1, 2, 3, and 4) for a standard solar model with normal composition and appoximately the correct age. It is found that theoretical eigenfrequencies calculated for our standard model agree approximately with observed peaks in the power spectra for the full-disk five minute oscillation of the Sun (Claverie et al., 1980; Grec et al., 1983; Scherrer et al., 1983) in agreement with other recent works (Christensen-Dalsgaard and Gough, 1980; Scuflaire et al., 1981). However, there still remains a slight discrepancy between theory and observations in such a sense that the theoretical eigenfrequencies are slightly lower than observations (see Figure 1).

An analysis of 28 contiguous days of whole disk observations of the solar surface by means of optical resonant scattering in the K 769.9 nm line, taken at the Teide Observatory at Izana during July-August 1980, have thus far yielded two significant facts. Firstly when the results of an iterative sine-wave fitting procedure are considered in the period range 2-3 h, although the expected daily harmonics corresponding to 1/8, 1/10, 1/11, and 1/12 of a day are clearly seen the l/9th contribution is significantly absent. It is suggested that this results from an interference between a signal of 160 min (l/9th of a day) with the daily harmonic. It is further pointed out that the observatories at which the 160 min oscillation has been seen, Crimea, Pic du Midi, and Stanford are all separated by integral numbers of 160 min, and thus the phase of the 160 min oscillation relative to the daily observation window is constant. However, the Teide Observatory is situated at a half integral number of 160 min periods relative to the others. Thus when constructive interference exist at the first three sites destructive interference will exist at the latter. It is thus concluded that the non-existence of a peak corresponding to the l/9th harmonic of a day in the sine-wave fit data is strong indirect evidence for the existence of the 160 min signal.

An analysis of those same data in the 5 min region has revealed the now well established pattern of discrete frequencies and with the increased resolution obtainable from 28 contiguous days of data, clearly showed the existence of splitting by rotational effects. In all 33 discrete lines were considered in the frequency range 2.4–3.85 mHz, which could be divided up into 3 groups, each of 11 lines, corresponding to the l = 0, l = 1, and l = 2 modes. This definitive classification was possible as the lines are split into (2l + 1) components yielding easily identifiable singlets, triplets and quintuplets. This first observation of the rotational splitting of solar oscillations gave the further information that the splitting of 0.75 ± 0.10 μHz indicated that the solar interior is rotating more rapidly than the observable surface (uniform rotation would yield a splitting of 0.4 μHz). A comparison of the widths of the individual peaks, with the intrinsic resolution of the data string ( ~ 1/T where T = length of the data string), showed that these were consistent with a high Q value oscillation, a fact which is further confirmed by the very existence of singlets triplets and quintuplets in the data.

The exact value of the speed of internal rotation of the Sun can only be deduced from these data by a model dependent calculation. The simplest of these would suggest that if the core were almost equal to the solar diameter then it is rotating twice as fast as the surface, whereas if the core were only 15 % of the solar diameter, it would be rotating at 9 times the surface rate.

The size of the rotational splitting recently observed (Claverie et al., 1981) is correlated with the 12.2d variation in the measurements of solar oblateness observed by Dicke (1976) and implies a convection zone of depth of 0.1 Rʘ. The near equality of amplitudes of global velocity oscillations (Claverie et al, 1981) of the various m components of the l = 1 and l = 2 modes as seen from the Earth viewing the Sun nearly along the equator is unexpected for pure rotational splitting. It is suggested that a magnetic perturbation is present and an oblique asymmetric magnetic rotator with magnetic fields of a few million gauss is responsible. A more detailed account was submitted to Nature.

A method of measurement of local line-of-sight velocities in the solar atmosphere by means of polarization optics is described. No spurious signals due to instrumental displacements of the spectrum arise with this method. The sensitivity of the method obtained is 0.3 m s-1, with a time constant τ = 5 s and input aperture 1.4ʺ x 4.5ʺ. Some preliminary results of the assessment of spatial characteristics of 5-min oscillations are included. Data are given to illustrate a center-to-limb variation of the spectrum of 5-min oscillations.

In this paper we describe a new observing system which is currently nearing completation at the Mount Wilson Observatory. This system has been designed to obtain daily measurements of solar photospheric and subphotospheric rotational velocities from the frequency splitting of non-radial solar p-mode oscillations of moderate to high degree (i.e. l > 150). The completed system will combine a 244 x 248 pixel CID camera with a high-speed floating point array processor, a 32-bit minicomputer, and a large-capacity disc storage system. We are integrating these components into the spectrograph of the 60-foot solar tower telescope at Mount Wilson in order to provide a facility which will be dedicated to the acquisition of oscillation data.

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 lowl modes to very small amplitudes. For g1, 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.

The coupling of non-radial pulsations and convection is studied on a simple example: an unstable semi-infinite polytrope. An expansion is proposed and the equations that should describe correctly the pulsations are isolated. The additional complications of the real solar case are discussed at the end.

We survey the mathematics of non-linear Hamiltonian oscillations with emphasis being laid on the more recently discovered Kolmogorov instability. In the context of radial adiabatic oscillations of stars this formalism predicts a Kolmogorov instability even at low oscillation energies, provided that sufficiently high linear asymptotic modes have been excited.

Numerical analysis confirms the occurrence of this instability. It is found to show up already among the lowest order modes, although high surface amplitudes are then required (ǀδrǀ/R ~ 0.5 for an unstable fundamental mode – first harmonic coupling). On the basis of numerical evidence we conjecture that in the Kolmogorov unstable regime the enhanced coupling due to internal resonance effects leads to an equipartition of energy over all interacting degrees of freedom. We also indicate that the power spectrum of such oscillations is expected to display two components: A very broad band of overlapping pseudo-linear frequency peaks spread out over the asymptotic range, and a strictly non-linear 1/f-noise type component close to the frequency origin.

It is finally argued that the Kolmogorov instability is likely to occur among non-linearly coupled non-radial stellar modes at a surface amplitude much lower than in the radial case. This lends support to the view that this instability might be operative among the solar oscillations.

Numerical solutions of the general time-dependent gas-dynamical equations in linear adiabatic approximation are given for initial conditions imitating: (a) a central perturbation, (b) a boundary perturbation (in the convective envelope), and (c) a ‘shrinking’ of the Sun as a whole. For a variety of models of the Sun it is found that at the surface the radial component vr of velocity is much greater than the tangential component vt, and that the period T of stationary oscillations does not exceed 131m. The appearance at the surface of a g mode with period 160m is found to be improbable.

With the initial conditions adopted, a propagating wave is produced which is reflected successively from the centre to the periphery and back, producing 5-min oscillations at the surface of the Sun. Expansion of this wave into separate modes leads to a power spectrum qualitatively similar to that observed.

The differential equations describing stellar oscillations are transformed into an algebraic eigenvalue problem. Frequencies of adiabatic oscillations are obtained as the eigenvalues of a banded real symmetric matrix. We employ the Cowling-approximation, i.e. neglect the Eulerian perturbation of the gravitational potential, and, in order to preserve selfadjointness, require that the Eulerian pressure perturbation vanishes at the outer boundary. For a solar model, comparison of first results with results obtained from a Henyey method shows that the matrix method is convenient, accurate, and fast.

Fitting the results of linear normal-mode analysis of the solar five-minute oscillations to the observed κ – ω diagram selects a class of models of the Sun’s envelope. It is a property of all the models in this class that their convection zones are too deep to permit substantial transmission of internal g modes of degree 20 or more. This is in apparent conflict with Hill and Caudell’s (1979) claim to have detected such modes in the photosphere.

A proposal to resolve the conflict was made by Rosenwald and Hill (1980). They pointed out that despite the impressive agreement between linearized theory and observation, nonlinear phenomena in the solar atmosphere might influence the eigenfrequencies considerably. In particular, they suggested that a correct nonlinear analysis could predict a shallow convection zone. This paper is an enquiry into whether their hypothesis is plausible.

We construct κ – ω diagrams assuming that the modes suffer local nonlinear distortions in the atmosphere that are insensitive to the amplitude of oscillation over the range of amplitudes that are observed. The effect of the nonlinearities on the eigenfrequencies is parameterized in a simple way. Taking a class of simple analytical models of the Sun's envelope, we compute the linear eigenfrequencies of one model and show that no other model can be found whose nonlinear eigenfrequencies agree with them. We show also that the nonlinear eigenfrequencies of a particular solar model with a shallow eonvective zone, computed with more realistic physics, cannot be made to agree with observation. We conclude, therefore, that the hypothesis of Rosenwald and Hill is unlikely to be correct.

A review is given of the properties of waves in atmospheres, with particular emphasis on (Section 1) the variation of amplitude and phase with altitude for propagating waves (Figures 1 to 4) and the waveforms of standing modes (Figure 5). The cases dealt with concern waves under the combined influences of gravity and compressibility, and examine the effects of: (Section 2) temperature gradients in a non-isothermal atmospheric model; (Section 3) external magnetic field, either vertical or horizontal; (Section 4) dissipation by viscosity and electrical resistance. The results are relevant to (Section 5) the assessment of atmospheric wave growth and shock formation, and to the calculation of heating functions describing the deposition of wave energy.

The present work deals with the theory of oscillations with periods of about 3 min observed in the chromosphere above sunspot umbrae. The model of these oscillations (slow mode magneto-acoustic waves trapped in a chromospheric resonant cavity) provides an independent method of checking empirical models of the chromosphere above sunspots.

Making use of this method, we investigate sunspot models which have been derived from spectroscopic data; the calculated periods of the oscillations fit well the observed periods.

Perturbation theory is developed for calculating the influence of slow differential rotation on the adiabatic nonradial modes of stellar oscillations. The effects of Coriolis forces and ellipticity are analysed simultaneously using the perturbation technique for Hermitian operators which is developed up to second order in eigenvalues and to first order in eigenvectors.

Modern development of particle physics makes it probable that new, still undiscovered, particles exist interacting with the ordinary matter by means of gravitation only. Okun suggested to call such particles the y-particles and to call the matter consisting of them the y-matter. We show that planet-like object orbiting inside the Sun and consisting of y-matter may explain 160 min nonradial solar oscillations.

Preliminary solar envelope models have been computed using the single-mode anelastic equations as a description of turbulent convection. This approach provides estimates for the variation with depth of the largest convective cellular flows, akin to giant cells, with horizontal sizes comparable to the total depth of the convection zone. These modal nonlinear treatments are capable of describing compressible motions occurring over many density scale heights. Single-mode anelastic solutions have been constructed for a solar envelope whose mean stratification is nearly adiabatic over most of its vertical extent because of the enthalpy (or convective) flux explicitly carried by the big cell; a sub-grid scale representation of turbulent heat transport is incorporated into the treatment near the surface. The single-mode equations admit two solutions for the same horizontal wavelength, and these are distinguished by the sense of the vertical velocity at the center of the three-dimensional cell. It is striking that the upward directed flows experience large pressure effects when they penetrate into regions where the vertical scale height has become small compared to their horizontal scale. The fluctuating pressure can modify the density fluctuations so that the sense of the buoyancy force is changed, with buoyancy braking actually achieved near the top of the convection zone. The pressure and buoyancy work in the shallow but unstable H+ and He+ ionization regions can serve to decelerate the vertical motions and deflect them laterally, leading to strong horizontal shearing motions. It appears that such dynamical processes may explain why the amplitudes of flows related to the largest scales of convection are so feeble in the solar atmosphere.

A procedure is outlined for estimating the influence of large-scale convective eddies on the wave patterns of five-minute oscillations of high degree. The method is applied to adiabatic oscillations, with frequency ω and wave number k, of a plane-parallel polytropic layer upon which is imposed a low-amplitude convective flow. The distortion to the k – ω relation has two constituents: one depends on the horizontal component of the convective velocity and has a sign which depends on the sign of ω/k; the other depends on temperature fluctuations and is independent of the sign of ω/k. The magnitude of the distortion is just at the limit of present observational sensitivity. Thus there is reasonable hope that it will be possible to reveal some aspects of the large-scale flow in the solar convection zone