STRUCTURE AND ROTATION OF THE SOLAR INTERIOR: INITIAL RESULTS FROM THE MDI MEDIUM-L PROGRAM

The medium-l program of the Michelson Doppler Imager instrument on board SOHO provides continuous observations of oscillation modes of angular degree, l, from 0 to 300. The data for the program are partly processed on board because only about 3% of MDI observations can be transmitted continuously to the ground. The on-board data processing, the main component of which is Gaussian-weighted binning, has been optimized to reduce the negative influence of spatial aliasing of the high-degree oscillation modes. The data processing is completed in a data analysis pipeline at the SOI Stanford Support Center to determine the mean multiplet frequencies and splitting coefficients. The initial results show that the noise in the medium-l oscillation power spectrum is substantially lower than in ground-based measurements. This enables us to detect lower amplitude modes and, thus, to extend the range of measured mode frequencies. This is important for inferring the Sun's internal structure and rotation. The MDI observations also reveal the asymmetry of oscillation spectral lines. The line asymmetries agree with the theory of mode excitation by acoustic sources localized in the upper convective boundary layer. The sound-speed profile inferred from the mean frequencies gives evidence for a sharp variation at the edge of the energy-generating core. The results also confirm the previous finding by the GONG (Gough et al., 1996) that, in a thin layer just beneath the convection zone, helium appears to be less abundant than predicted by theory. Inverting the multiplet frequency splittings from MDI, we detect significant rotational shear in this thin layer. This layer is likely to be the place where the solar dynamo operates. In order to understand how the Sun works, it is extremely important to observe the evolution of this transition layer throughout the 11-year activity cycle.     

On the effect of magnetic field on the low-l solar p-mode oscillations

One of the possible magnetic field effects on the stellar pulsations is known to be a splitting in the observed frequencies. Using this knowledge in the solar convection zone, there are two aims in this work Considering the Sun as an incompressible fluid, our first objective was to investigate the variation of the physical parameters in the 30% outermost convective solar layer, during a pulsation period. The second purpose was to calculate, by means of the spherical harmonics, the shifts on the low-l p-mode frequencies which could be caused by the presence of the magnetic field in the Sun. The first order perturbation approximation was used in order to calculate analytically the resulting frequency shifts and the small perturbations on the magnetic field, as well as the physical parameters, such as density, pressure and temperature, of a Standard Solar Model excluding both rotation and magnetic field (Christensen-Dalsgaard et al., 1996) in the unperturbed equilibrium case.     

Influence of Low-Degree High-Order p-Mode Splittings on the Solar Rotation Profile

The solar rotation profile is well constrained down to about 0.25R _⊙ thanks to the study of acoustic modes. Since the radius of the inner turning point of a resonant acoustic mode is inversely proportional to the ratio of its frequency to its degree, only the low-degree p modes reach the core. The higher the order of these modes, the deeper they penetrate into the Sun and thus they carry more diagnostic information on the inner regions. Unfortunately, the estimates of frequency splittings at high frequency from Sun-as-a-star measurements have higher observational errors because of mode blending, resulting in weaker constraints on the rotation profile in the inner core. Therefore inversions for the solar internal rotation use only modes below 2.4 mHz for ?≤3. In the work presented here, we used an 11.5-year-long time series to compute the rotational frequency splittings for modes ?≤3 using velocities measured with the GOLF instrument. We carried out a theoretical study of the influence of the low-degree modes in the region from 2 to 3.5 mHz on the inferred rotation profile as a function of their error bars.     

Is the solar core rotating faster or slower than the envelope?

The Sun is not a rigid body and it is well known that its surface rotation is differential, the polar regions rotating substantially slower than the equator. This differential rotation has been demonstrated by helioseismology to continue down to the base of the convective zone, below which it becomes closer to a rigid body rotation. Far deeper, inside the energy generating core, the rotation has generally been assumed to be much faster, keeping memory of the presumably high speed of the young Sun. However, several recent results of helioseismology have decreased this likelihood more and more, so that the core rotation could be suspected to be only marginally, or even not at all faster than the envelope. Certain results would even imply a core rotation slower than the envelope, an interesting but unlikely possibility. We present here a complete analysis of the rotational splitting of the low degree modes measured in three different time series obtained in 1990, 1991, and 1992 by the IRIS full-disk network. With a time of integration slightly longer than 4 months, the splitting has been measured by 4 different global methods on 42 doublets of l = 1, 35 triplets of l = 2, and 30 quadruplets of l = 3. With a high level of confidence, our result is consistent with a rigid solar core rotation.     

Evidence for a small,high-Z,iron-like solar core

The high-Z core (HZC) model of the Sun, supported in Rouse (1985) by superior agreements of nonradial g-mode periods of oscillation with long period observations, is used to calculate frequencies of oscillation in the five-minute band (5MB). Allowing for the fact that the present HZC model profile does not include an upper photosphere and self-consistent chromosphere, the HZC model of the Sun is also supported by the very good agreements of the 5MB nonradial frequencies of oscillation with observations for HZC l degrees 0 to 19 and orders n 20, and the good agreement of the HZC purely radial frequencies of oscillation with about the same n-orders with observations previously identified as l = 0 oscillations. Two important aspects of these agreements are (1) the nonradial frequencies were calculated with the equations that neglect the gravitational perturbation (the Cowling approximation), and (2) the radial frequencies were calculated with the equation that includes the gravitational perturbation. The present agreements suggest that for solar-type stars, the gravitational perturbation may not affect the nonradial p-modes of oscillation as much as it affects the radial modes and the nonradial g-modes. More research will be performed.     

Periodic solar EUV flux monitored near Venus

A detector sharing the orbital rate of Venus has a unique perspective on solar periodicities. Fourier analysis of the 8.6 year record of solar EUV output gathered by the Langmuir probe on Pioneer Venus Orbiter shows the influences of global oscillation modes located in the convective envelope and in the radiative interior. Seven of the eight lowest angular harmonic r-mode families are detected by their rotation rates which differ almost unmeasurably from ideal theoretical values. This determines a mean sidereal rotation rate for the envelope of 457.9 ± 2.0 nHz which corresponds to a period of 25.3 days. Many frequencies are aliased at ± 106 nHz by modulation from the lowest angular harmonic r-mode in the envelope. The rotation of this mode seems slightly retrograde, -1.5 ± 2.0 nHz, but small positive values are not excluded. We confirm that the rotation of the radiative interior, 381 nHz, is slower than the envelope by detecting g-mode frequencies for angular harmonics, 2 l 6, and a possible first detection of the rotation rate for the l = 1 case. Solar EUV lacks the sudden darkenings (dips) shown by visible irradiance; vortex cores in the photosphere and below are again suggested as a possible explanation.     

Rotation Profiles of Solar-like Stars with Magnetic Fields

We investigate the rotation profile of solar-like stars with magnetic fields. A diffu-sion coefficient of magnetic angular momentum transport is deduced. Rotating stellar models with different mass incorporating the coefficient are computed to give the rotation profiles. The total angular momentum of a solar model with only hydrodynamic instabilities is about 13 times larger than that of the Sun at the age of the Sun, and this model can not reproduce quasi-solid rotation in the radiative region. However, the solar model with magnetic fields not only can reproduce an almost uniform rotation in the radiative region, but also a total angular momentum that is consistent with the helioseismic result at the 3 σ level at the age of the Sun. The rotation of solar-like stars with magnetic fields is almost uniform in the radiative region, but for models of 1.2-1.5 M⊙, there is an obvious transition region between the convective core and the radiative region, where angular velocity has a sharp radial gradient, which is different from the rotation profile of the Sun and of massive stars with magnetic fields. The change of angular velocity in the transition region increases with increasing age and mass.     

Optimal Masks for Low-Degree Solar Acoustic Modes

We suggest a solution to an important problem in observational helioseismology of the separation of lines of solar acoustic (p) modes of low angular degree in oscillation power spectra by constructing optimal masks for Doppler images of the Sun. Accurate measurements of oscillation frequencies of low-degree modes are essential for the determination of the structure and rotation of the solar core. However, these measurements for a particular mode are often affected by leakage of other p-modes arising when the Doppler images are projected on to spherical harmonic masks. The leakage results in overlapping peaks corresponding to different oscillation modes in the power spectra. In this Letter, we present a method for calculating optimal masks for a given (target) mode by minimizing the signals of other modes appearing in its vicinity. We apply this method to time series of 2 yr obtained from the Michelson Doppler Imager instrument on board the Solar and Heliospheric Observatory space mission and demonstrate its ability to reduce efficiently the mode leakage.     

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.     

Analysis of MDI High-Degree Mode Frequencies and their Rotational Splittings

We present a detailed analysis of solar acoustic mode frequencies and their rotational splittings for modes with degree up to 900. They were obtained by applying spherical harmonic decomposition to full-disk solar images observed by the Michelson Doppler Imager onboard the Solar and Heliospheric Observatory spacecraft. Global helioseismology analysis of high-degree modes is complicated by the fact that the individual modes cannot be isolated, which has limited so far the use of high-degree data for structure inversion of the near-surface layers (r>0.97R _⊙). In this work, we took great care to recover the actual mode characteristics using a physically motivated model which included a complete leakage matrix. We included in our analysis the following instrumental characteristics: the correct instantaneous image scale, the radial and non-radial image distortions, the effective position angle of the solar rotation axis, and a correction to the Carrington elements. We also present variations of the mode frequencies caused by the solar activity cycle. We have analyzed seven observational periods from 1999 to 2005 and correlated their frequency shift with four different solar indices. The frequency shift scaled by the relative mode inertia is a function of frequency alone and follows a simple power law, where the exponent obtained for the p modes is twice the value obtained for the f modes. The different solar indices present the same result.     

Helioseismic solar cycle changes and splitting coefficients

Using the GONG data for a period over four years, we have studied the variation of frequencies and splitting coefficients with solar cycle. Frequencies and even-order coefficients are found to change significantly with rising phase of the solar cycle. We also find temporal variations in the rotation rate near the solar surface.     

Spherical harmonic analysis of steady photospheric flows

Steady photospheric flows can be represented by a spectrum of spherical harmonic modes. A technique is described in which full disc doppler velocity measurements are analysed using the spherical harmonic functions to determine the characteristics of this spectrum and the nature of these flows. Synthetic data is constructed for testing this technique. This data contains limb shift, rotation, differential rotation, meridional circulation, supergranules, giant cells and various levels of noise.The data is analysed in several steps. First, the limb shift is calculated by finding the average velocity in concentric rings about disc center. A polynomial representation of the limb shift is then removed from the data. Secondly, the rotation profile is calculated by finding an average slope in the velocity across the disc at each latitude position. This rotation profile is fit with Legendre polynomials and removed from the data. The third step is to find the meridional circulation by calculating the spherical harmonic transform for the axisymmetric poloidal modes and correcting for the effects of the limb shift analysis. The final step is to calculate the full spectrum of spherical harmonic components for the convective flows. Supergranules are separated from giant cells by spectral filtering for high (l >32) and low (l <32) wavenumbers, respectively.Some information about the spectrum is lost because only one hemisphere is seen, only the line-of-sight velocity is measured and the measurements contain noise. The lack of information about the motions on the backside of the Sun produces a broad smearing of the spectrum into nearby modes. The lack of information about the transverse velocity component produces a mixing between modes whose longitudinal wavenumbers differ by two and between the poloidal and toroidal components with the same wavenumber. In spite of this mode mixing much can be learned from this analysis. Solar rotation and differential rotation can be accurately measured and monitored for secular changes. Meridional circulations with small amplitudes can be measured and monitored and giant cells can be separated from supergranules.     

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.     

Helioseismic search for magnetic field in the solar interior

The observed splittings of solar oscillation frequencies can be utilized to study possible large-scale magnetic fields present in the solar interior. Using the GONG data on frequency splittings an attempt is made to infer the strength of magnetic fields inside the Sun.     

The subsurface radial gradient of solar angular velocity from MDI f-mode observations

We report quantitative analysis of the radial gradient of solar angular velocity at depths down to about 15 Mm below the solar surface for latitudes up to 75° using the Michelson Doppler Imager (MDI) observations of surface gravity waves (fmodes) from the Solar and Heliospheric Observatory (SOHO). A negative outward gradient of around –400 nHz/R , equivalent to a logarithmic gradient of the rotation frequency with respect to radius which is very close to –1, is found to be remarkably constant between the equator and 30° latitude. Above 30° it decreases in absolute magnitude to a very small value at around 50°. At higher latitudes the gradient may reverse its sign: if so, this reversal takes place in a thin layer extending only 5 Mm beneath the visible surface, as evidenced by the most superficial modes (with degrees l>250). The signature of the torsional oscillations is seen in this layer, but no other significant temporal variations of the gradient and value of the rotation rate there are found.     

Differential rotation and meridional circulation in global models of solar convection

In the outer envelope of the Sun and in other stars, differential rotation and meridional circulation are maintained via the redistribution of momentum and energy by convective motions. In order to properly capture such processes in a numerical model, the correct spherical geometry is essential. In this paper I review recent insights into the maintenance of mean flows in the solar interior obtained from high-resolution simulations of solar convection in rotating spherical shells. The Coriolis force induces a Reynolds stress which transports angular momentum equatorward and also yields latitudinal variations in the convective heat flux. Meridional circulations induced by baroclinicity and rotational shear further redistribute angular momentum and alter the mean stratification. This gives rise to a complex nonlinear interplay between turbulent convection, differential rotation, meridional circulation, and the mean specific entropy profile. I will describe how this drama plays out in our simulations as well as in solar and stellar convection zones. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Low-degree solar oscillation spectrum with wavelet decomposition

We present the technique and results of processing the observations of the solar oscillations with frequencies of 1.5–4.0 mHz. We used the time series obtained in the period 1990–1996 in the International IRIS project. The power spectrum averaged over the entire observing period was estimated from the spectra of each year preaveraged with five orthogonal tapers. The spectral profiles of the individual modes with degree l = 0 and orders n ? 12–26 were fitted using a wavelet decomposition with thresholding of the high-frequency decomposition levels. The smoothed power spectrum estimate has a considerably smaller variance, but also a lower density of reference points. The derived smoothed profiles were used to determine the eigenfrequencies by two methods: from the local peak and from the correlation between the neighboring peaks, which correspond to a nonparametric estimation procedure. For the radial modes (l = 0), there is good agreement with the independent parametric maximum likelihood frequency estimates.     

Why the Sun may appear oblate

If the Sun loses angular momentum from its core, due to core contraction, into the solar wind at the observed rate, then an 0.7 day rotational period for the core of the Sun is required for temporal equilibrium. The rotational power released in the core contraction process can equal the observed magnetic energy released in the solar activity cycle if the Sun's core rotates with a period near 1.4 to 4 days. The rotational power released from a rotating object is , where is the torque on the object and is its angular velocity. Fitting this to the solar wind torque and core rotation rate provides an 0.5 to 5 day rotation period for the Sun's core. A gravitational Pannekoek-Rosseland electric field in the Sun makes the Ferraro theorem inapplicable in such a way that rather than a constant angular velocity with radius, an inverse square radial dependence occurs. This results in a two day rotational period for the region in the Sun where most of the angular momentum resides. The consistency of the above four methods suggests that the Sun's observed oblateness is due to a rapidly rotating solar core. The oblateness of the photosphere is estimated to be near 3.4×10^–5.     

The Dynamics of the Solar Radiative Zone

The dynamics of the solar radiative interior are still poorly constrained by comparison to the convective zone. This disparity is even more marked when we attempt to derive meaningful temporal variations. Many data sets contain a small number of modes that are sensitive to the inner layers of the Sun, but we found that the estimates of their uncertainties are often inaccurate. As a result, these data sets allow us to obtain, at best, a low-resolution estimate of the solar-core rotation rate down to approximately 0.2R _⊙. We present inferences based on mode determination resulting from an alternate peak-fitting methodology aimed at increasing the amount of observed modes that are sensitive to the radiative zone, while special care was taken in the determination of their uncertainties. This methodology has been applied to MDI and GONG data, for the whole Solar Cycle 23, and to the newly available HMI data. The numerical inversions of all these data sets result in the best inferences to date of the rotation in the radiative region. These results and the method used to obtain them are discussed. The resulting profiles are shown and analyzed, and the significance of the detected changes is discussed.     

Properties of r-modes in the Sun

Global oscillations of the Sun (r-modes) with very long periods 1 month are reviewed and studied. Such modes would be trapped in an acoustic cavity formed either by most of the convective envelope or by most of the radiative interior. A turning point frequency giving cavity boundaries is defined and the run of eigenvalues for angular harmonics l 3 are plotted for a conventional solar convection zone. The r-modes show equipartition of oscillatory energy among shells which each contain one antinode in the radial dimension. Toroidal motion is dominant to at least the 14^th radial harmonic mode. Viscosity from convective turbulence is strong and would damp any mode in just a few solar rotations if it were the only significant nonadiabatic effect. Radial fine splitting which lifts the degeneracy in n is very small (20 nHz or less) for all n 14 trapped in the envelope. But, if splitting could be detected, we would have a valuable new constraint on solar convection theories.