The following is a somewhat condensed version of discussions previously given elsewhere (Spruit, 1991, 1992). Some new developments not discussed there are presented in sections 4 and 5.

Since the observed irradiance variations are so clearly associated with manifestations of the solar magnetic field, I focus here on magnetic causes. Much of the physics of irradiance variations, however, is governed by the thermal response of the convective envelope and this response is similar for other possible causes of irradiance variations.

Kuhn et al. (1988) have found that there are variations in the photospheric temperature with the solar cycle that depend on solar latitude. This would be an independent mechanism, other than the effects of sunspots and faculae, contributing to the change in solar irradiance (Kuhn 1991). It is important to pursue this investigation as such variations would be related to the transport of energy through the convection zone, and thus give a good indication of its structure and evolution.

The radial order solar p-modes show an amplitude modulation at the equatorial siderial rotation frequency of the Sun. The peak is clearly separated from the peak at the synodic rotation visible in the irradiance data.

Based on detailed 2D numerical radiation hydrodynamics (RHD) calculations of time-dependent compressible convection, we have studied the dynamics and thermal structure of the convective surface layers of stars in the range of effective temperatures and gravities between 4500 K ≤ Teff ≤ 7100 K and 2.54 ≤ log g ≤ 4.74. Although our hydrodynamical models describe only the shallow, strongly superadiabatic layers at the top of the convective stellar envelope, they provide information about the value of the entropy s∗ of the deeper, adiabatically stratified regions. E.g. in the solar case the helioseismically measured entropy jump is predicted within 9% of its actual value.

The technique which we use is based on the “quasi-asymptotic” description of solar p-modes developed by Roxburgh & Vorontsov (1996). As a significant improvement of the standard asymptotic analysis, this description incorporates a Born approximation to allow for a detailed treatment of the regions of rapid variation of seismic parameters with depth (base of the convection zone) and strong influence of gravity perturbations (low-degree modes which penetrate deep into the solar core).

There are many investigations of the total solar irradiance (TSI) variability over the time scales from days to years. Particular emphasis has been placed on the sources of this variability and on the problems concerning stable equipment operation. The investigations made aboard SMM/ACRIM showed a relation between solar variability and various solar activity manifestations. Even then the TSI power spectrum revealed some peculiar periodicity in the range from 9 to 11 days that was not explained and also a well-defined quasi-4-day periodicity (Froelich and Pap, 1989). Their distinctive feature is that they manifest themselves in many processes both on Sun and on Earth. An understanding of their nature is of importance to solar-terrestrial physics. The earlier investigations of Nimbus-7 data pointed out the possible effect of solar sources of unknown nature and also some residual instrumental effects on the behaviour of TSI variations. As a further development of these studies the spectral analysis was made of the TSI variations using the final data set derived over the range from November 1978 to April 1992 aboard Nimbus-7. In our study the data gaps were filled with an artificial noise signal. The autocorrelation function (ACF) was used for detrending and filtering of the data.

One region where discrepancies between the seismic and standard solar models continue to exist is the overshoot layer below the convective zone (Gough et al. 1996). A main reason for this discrepancy is clearly the lack of a realistic physical model for astrophysical overshoot. Beside numerical experiments, the most promising approach to the construction of such a model is based on the Reynolds-stress formalism of turbulence theory (Speziale 1991). As a first step toward the development of a more complete Reynolds stress model of the overshoot, here we present a simple κ-ϵ model, valid under the following conditions:

1. (a). Incompressibility: ∂iVi = 0. Owing to the short length scale (cf. Fig. 1), this assumption is somewhat better founded here than in other models.

2. (b). Closure: Let and . (w = vz vertical velocity; κ ≡ V2/2: kinetic energy; ϵ: dissipation rate). Now assume Cwk = const., Cwt =const. This is warranted if l/Hv ≃constant (Hv: scale height of V2), which will be seen indeed to be the case in our model, cf. eqs. (3)–(4).

3. (c). No correlation with thermodynamic fluctuations: . The assumption implies that these correlations vanish in a very thin (≪ Hp) transition layer below △▽ = 0. No change in the geometrical flow properties is expected in such a thin layer, so lo, Cwk, and Cwϵ should be the same as in the lower part of the unstable layer. On the basis of numerical experiments here we use lo = HP, Cwk = 0.6, Cwϵ = 3, q = 2.4. (The index ‘0’ refers to values at z = 0 i.e. at △▽ = 0.) Note however that V may be significantly reduced in the transition layer, so Vo (and thus ) is undetermined in the present model.

In continuation of our earlier studies (Singh, Roxburgh & Chan 1997a,b; hereafter SRC97a,b) we perform some further tests to study the general behaviour of penetrative convection and its scaling with rms vertical velocity by varying a number of input parameters like the aspect ratio and the positioning of the interface between the unstable-lower stable layer.

I want to start by addressing the question, ‘What is inversion?’ My answer would be that inversion is the process of going from data to making inferences about the object under study. In the case of helioseismology, the data at the present time are principally the mode frequencies, and the object under study is the solar interior.

The study of the internal structure of the Sun entered a new age with helioseismology. Several ground based networks, as well as the space mission SoHO, continuously observe the solar oscillations. In particular GONG, the “Global Oscillation Network Project”, gathers six sites around the world with six identical Doppler instruments. These instruments observe the phase shift of the Ni 676.8nm line with 3 images every minute, 1.8 sites observing simultaneously. Millions of solar p-modes have been detected. The inversion of the measured frequencies yields accurate and detailed information about the sound velocity in the Sun's interior, which in turn leads to constraints on the equation of state, opacities, chemical composition. Precise informations on the differential rotation inside the Sun have also been obtained. (See the special Science issue on GONG, vol 272, 31 May 1996).

With the flood of high quality helioseismic data from the instruments on the SOHO spacecraft (MDI/VIRGO/GOLF) and ground based instruments (e.g. GONG, LOWL, BiSON, IRIS, MWO-CrAO and TON) we have been able to get increasingly detailed information on the rotation and other large scale flows in the solar interior.

Here we will discuss some of the highlights of what we have learned so far and what we may expect to learn in the near future. Among the recent advances have been tighter constraints on the tachocline at the bottom of the convection zone and detection of details in the surface rotation rate similar to the torsional oscillations found in the surface Doppler shift.

The technique of time-distance helioseismology provides a new tool for examining large-scale flows beneath the sun's surface. We have used this technique to study the meridional flow and have found that the poleward flow observed at the surface appears to persist to a depth of at least 26,000 km. We have also examined the differential rotation in these uppermost layers, and find a slight asymmetry between the northern and southern hemispheres.

We used GONG time series for ℓ = 0,1,2,3 from June 10, 1995 to January 7, 1997, 578 days in total. The duty cycle varies significantly during this period of time but on average it is sufficiently high, and the remaining gaps are irregular. Because of this we do not see daily side lobes in the spectra obtained.

We report the initial results of one year of continuous observations of the Sun‘s internal structure from the Michelson Doppler Imager (MDI) on board SOHO. The results have been obtained by inverting frequencies of p and f modes determined with two different methods of averaging over split multiplets. Small systematic differences between the two frequency sets depend primarily on mode frequencies, and, thus, did not significantly affect the inversions. A preliminary study of the systematic effects resulting from asymmetry of oscillation power peaks has also shown no significant influence on the inversion results. The inferred sound-speed profile is in general agreement with the previous data from MDI and ground-based networks. In the energy-generating core, the resolution is substantially improved, and the inversion results indicate a sharp negative perturbation of the sound speed in the core, tending to a positive value near the center. High-precision measurements of the f-mode frequencies have been used to determine the seismic radius of the Sun. The global asphericity estimated from frequency variation across the split mode multiplets has been found to be small, and is consistent with the asphericity during the previous activity minimum. Variations of the solar frequencies during the first year of MDI observations have also been detected.

The fundamental mode (f-mode) which is essentially a surface mode can be expected to provide a diagnostic of flows and magnetic fields etc. present in the surface regions (Ghosh, Chitre & Antia 1995; Rosenthal & Christensen-Dalsgaard 1995). It turns out that the observed f-mode frequencies can also provide an accurate measure of solar radius (Antia 1997; Schou et al. 1997).

Two different data sets have been used to derive low-degree rotational splittings. One data set comes from the Luminosity Oscillations Imager of VIRGO on board SOHO; the observation starts on 27 March 96 and ends on 26 March 97, and are made of intensity time series of 12 pixels (Appourchaux et al, 1997, Sol. Phys., 170, 27). The other data set was kindly made available by the GONG project; the observation starts on 26 August 1995 and ends on 21 August 1996, and are made of complex Fourier spectra of velocity time series for l = 0 − 9. For the GONG data, the contamination of l = 1 from the spatial aliases of l = 6 and l = 9 required some cleaning. To achieve this, we applied the inverse of the leakage matrix of l = 1, 6 and 9 to the original Fourier spectra of the same degrees; cleaning of all 3 degrees was achieved simultaneously (Appourchaux and Gizon, 1997, these proceedings).

An in-depth discussion of the analysis presented here can be found in an up-coming paper (Chaplin et al., 1997).

The presence of a magnetic field will raise the degeneracy in ℓ of the resonant p-mode oscillations, via perturbations resulting from the Lorentz force. These degeneracyraising effects will give rise to asymmetric mode-multiplet structures. Both Gough & Thompson (1990). and Dziembowski & Goode (1997) have addressed the implications and potential complications that might result from such phenomena. Here, in an attempt to reveal the presence of an asymmetric frequency structure in the low-degree ℓ = 2 modes, i.e. to measure the asymmetries

An in-depth discussion of the analysis presented here can be found in an up-coming paper (Chaplin et al., 1997).

In order to investigate the solar-cycle dependence of the low-degree p-mode frequencies, we have analysed eighteen 4-month frequency spectra generated from BiSON Doppler velocity residuals collected between 1991 January 01 and 1997 January 05. These data cover the falling phase of solar activity cycle 22. up to the cycle 22/23 boundary. Fig. 1(a) shows frequency shifts, averaged over two orders in n. up to ~ 3900 μHz - as derived from the analysis of the 4-month spectra - normalised to unit change in the 10.7-cm radio flux. The dashed line lying above the BiSON data is a fit to the BBSO 1989-minus-1986 frequency shifts, for 4 ≤ ℓ ≤ 140 (Libbrecht k Woodard. 1991). The dotted line passing through the data corresponds to the bestscaled fit of the BBSO data to the BiSON data - the best-fit requires the BBSO data to be scaled by 0.71 ± 0.03. This is reasonably consistent - as expected - with the mean, overall ratio of the inverse mode masses of those data used in the BiSON and BBSO analyses (≈ 0.67). Fig. 1(b) shows the BiSON activity-normalised frequency shifts, plotted as a function of inverse mode mass - here, as anticipated, there is a clear correlation between the variables.

The systematic p-mode frequency changes which are observed through the solar cycle are believed to be associated with near-surface perturbations confined to the magnetically active latitudes. In this paper, we study the perturbation arising from the presence of a large “active region”, corresponding to a localized structural change in a thin region close to the photosphere. We shall ignore the difficult question of the magnitude of the effect, and simply consider some geometric and observational implications for low-degree modes.

The Ring Diagram analysis is a technique designed to infer the presence of horizontal velocity flows under the solar surface by the analysis of the power spectra of solar oscillations in three dimensions: the two components of the horizontal wave number and temporal frequency (Hill, 1988 and Patrón et al., 1995). This procedure is applied as a local analysis technique, performed for a region of the Sun of several heliographic degrees (typically around 15×15 degrees square). As the Sun is rotating, we must track a chosen region as long as it is present at the visible part of the solar disk, and we can continue the tracking after several solar rotations. The comparison of the estimated flow fields obtained at different times can give some ideas about the temporal variations of the flows.