On comparing helioseismic two-dimensional inversion methods

Well-confirmed helioseismic data from several groups using various observational techniques at different sites have allowed us to determine the differential rotation in the outer half of the Sun's interior. The resulting rotation law is simple — the surface differential rotation persists through much of the convection zone with a transition toward solid body rotation beneath. To date there is no appealing evidence for a rapidly rotating core. There is however, weak evidence for a solar cycle dependence of the Sun's internal rotation.

It is well known that the combination of barotropic rotation and radiative equilibrium are mutually incompatible in stars. The Sun's internal rotation is far from barotropic, however, which allows at least the theoretical possibility that the Sun's thermal balance is one of radiative equilibrium in the region of the tachocline near the outer boundary of the radiative zone. We show here that (i) the constraint of radiative equilibrium leads to a straightforward ordinary differential equation for the Sun's rotation profile, and (ii) solutions of this equation can be found that, to within current levels of accuracy, closely resemble the rotation profile deduced from helioseismology. More generally, we calculate how large a baroclinic deviation from uniform rotation is required to maintain radiative equilibrium without meridional circulation throughout the bulk of the radiative zone. Very little deviation is required, well below detectability. The feasibility of radiative equilibrium for the tachocline suggests that the issue of a spreading boundary layer may be less severe than previously thought.

We report inferences for the Sun's internal rotation from GONG months 4-10 averaged power spectra. In keeping with the international collaborative nature of the GONG project, the results presented here are based on the work of several groups around the world inverting the GONG data and sharing their results via the world-wide web. These groups are at the Observatoire de la Côte d'Azur, Nice (T. Corbard, G. Berthomieu, J. Provost); Theoretical Astrophysics Center, Aarhus (J. Christensen-Dalsgaard, F. Pijpers); Center for Astrophysics, Cambridge MA (A. Eff-Darwich, S. Korzennik); QMW, London (R. Howe, M. Thompson, in collaboration with J. Schou, Stanford); Institute of Astronomy, Cambridge (T. Sekii, D. Gough); University of Sydney (D. Burtonclay, Li Yan, P. Wilson); and Tata Institute of Fundamental Research, Bombay (H. Antia, S. Chitre).

We examine the linear behavior of three-dimensional Lagrangian displacements in a stratified, shearing background. The isentropic and iso-rotation surfaces of the equilibrium flow are assumed to be axisymmetric, but otherwise fully two-dimensional. Three-dimensional magnetic fields are included in the perturbation equations; however the equilibrium is assumed to be well-described by purely hydrodynamic forces. The model, in principle very general, is used to study the behavior of fluid displacements in an environment resembling the solar convection zone. Some very suggestive results emerge. All but high-latitude displacements align themselves with the observed surfaces of constant angular velocity. The tendency for the angular velocity to remain constant with depth in the bulk of the convective zone, together with other critical features of the rotation profile, emerge from little more than a visual inspection of the governing equation. In the absence of a background axial angular velocity gradient, displacements exhibit no poleward bias, suggesting that solar convection "plays-off" of prexisting shear rather than creates it. We argue that baroclinic vorticity of precisely the right order is generated at the radiative/convective zone boundary due to centrifugal distortion of equipotential surfaces that is not precisely followed by isothermal surfaces. If so, many features of the Sun's internal rotation become more clear, including: i) the general appearance of the tachocline; ii) the extension of differential rotation well into the radiative zone; iii) the abrupt change of morphology of convective zone isorotation surfaces; and iv) the inability of current numerical simulations to reproduce the solar rotation profile without imposed entropy boundary conditions.

An introduction to current techniques to infer the Sun's internal rotation from observed acoustic oscillation modes is given, and some representative results are shown.

Sound waves generated in the solar convection zone propagate through the Sun and cause it to vibrate. Resonant vibrations are detected by measuring wavelike motion that they produce on the solar surface; by analyzing them we can learn about conditions in the solar interior. At present we have only limited data but already we have drawn an important conclusion. We have inferred that outside the core of the Sun, and beneath the subsurface convection zone, the abundance of helium is about 25% by mass. It is generally believed that the intermediate region in the Sun is uncontaminated by the products of the nuclear reactions in the core and by material that might have fallen into the Sun from interstellar space. If that is indeed the case, we have measured the helium abundance of the gas cloud from which the Sun condensed some 4.7 thousand million years ago. This information is relevant to our understanding of the early stages of the evolution of the Universe. It is exciting that there are hints in the vibration data that might even enable us to determine whether the Sun really has remained unmixed, and hence whether we have a faithful record of the primordial mixture. There is also evidence in the data concerning the Sun's internal rotation. It appears that the core of the Sun rotate faster than the surface, but by how much is a subject of some controversy. It is very important to settle the matter, because the result is an essential step in one of the most important tests of Einstein's General Theory of Relativity. What evidence we have at present does not contradict Einstein's theory.

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.

In this work we present the extended two-dimensional torsion (E2DT) method and use it to analyze the performance of several methods that incorporate torsional anharmonicity more approximately for calculating rotational-vibrational partition functions. Twenty molecules having two hindered rotors were studied for temperatures between 100 and 2500 K. These molecules present several kinds of situations; they include molecules with nearly separable rotors, molecules in which the reduced moments of inertia change substantially with the internal rotation, and molecules presenting compound rotation. Partition functions obtained by the rigid-rotor harmonic oscillator approximation, a method involving global separability of torsions and the multistructural methods without explicit potential coupling [MS-T(U)] and with explicit potential coupling [MS-T(C)] of torsions, are compared to those obtained with a quantized version - called the extended two-dimensional torsion (E2DT) method - of the extended hindered rotor approximation of Vansteenkiste et al. ( Vansteenkiste et al. J. Chem. Phys. 2006 , 124 , 044314 ). In the E2DT method, quantum effects due to the torsional modes were incorporated by the two-dimensional nonseparable method, which is a method that is based on the solution of the torsional Schrödinger equation and that includes full coupling in both the kinetic and potential energy. By comparing other methods to the E2DT method and to experimental thermochemical data, this study concludes that the harmonic approximation yields very poor results at high temperatures; the global separation of torsions from the rest of the degrees of freedom is not justified even when an accurate method to treat the torsions is employed; it is confirmed that methods based on less complete potential energy coupling of torsions, such as MS-T(U), are not accurate when dealing with rotors with different barrier heights, and more complete inclusion of torsional coupling to the method in MS-T(C) improves substantially the results in such a way that it could be used in cases where the E2DT method is unaffordable.

Context. Open clusters provide unambiguous clues to understand the evolution of 7Li at the surface of low-mass stars and its possible correlation with stellar rotation, which is a challenge for both stellar hydrodynamics and Galactic chemical evolution. Aims. We aim to quantify the efficiency of the transport processes for both angular momentum and chemicals that are required to explain simultaneously the observed behaviour of surface 7Li (and 9Be) and rotation as well as the internal rotation profiles inferred from helio- and asteroseismology in F- and G-type main sequence stars. Methods. We apply the model for the transport of angular momentum and chemicals that we tailored in a previous work for solar-type stars to an extended range of initial masses and metallicities corresponding to F- an G-type stars in a sample of 20 Galactic open clusters. We evaluate its ability to explain the 7Li, 9Be, and rotation periods observations. This model includes atomic diffusion, rotation-induced processes (for which we tested different prescriptions for shear turbulence), penetrative convection with a rotational dependence, parametric viscosity and turbulence, and magnetic braking. Results. Over the entire range of masses, metallicities, and ages explored, we reproduce the evolution of the surface rotation rates and predict, for the first time, the observed anti-correlation between the surface rotation rate and 7Li depletion as a consequence of the penetrative convection prescription. The 7Li behaviour and its evolution with time is well reproduced for G-type stars. However, the ability of the model to reproduce the so-called 7Li dip centred around ∼6600 K strongly depends on the adopted prescriptions for shear turbulence. It also requires a stellar mass dependence for the parametric viscosity adopted for the transport of angular momentum, similar to the behaviour predicted for the generation and luminosity of internal gravity waves generated by stellar convective envelopes. Finally, the model predicts internal rotation profiles in good agreement with asteroseismic constraints in main sequence stars. Conclusions. We provide an efficient way to model G-type stars of different ages and metallicities successfully. However, the 7Li and 9Be dip constraints urgently call for further hydrodynamical studies to better model turbulence in stars, and for the exploration of physical processes such as tachocline mixing for the transport of chemicals and internal gravity waves for the transport of angular momentum. Finally, additional data for the internal rotation and for 9Be in main sequence low-mass stars are definitively needed.

The potential of a proposed spacecraft mission, called Starprobe, for testing general relativity and providing information on the interior structure and dynamics of the sun is investigated. Parametric, gravitational perturbation terms are derived which represent relativistic effects and effects due to spatial and temporal variations in the solar potential at a given radial distance. A covariance analysis based on Kalman filtering theory predicts the accuracies with which the free parameters in the perturbation terms can be estimated with radio metric tracking data through the process of trajectory reconstruction. It is concluded that Starprobe can contribute significant information on both the nature of gravitation and the structure and dynamics of the solar interior.

Understanding Jupiter's present‐day interior structure and dynamics is key to constraining planetary accretion models. In particular, the extent of stable stratification (i.e., non‐convective regions) in the planet strongly influences long‐term cooling processes, and may record primordial heavy element gradients from early in a planet's formation. Because the Galileo entry probe measured a subsolar helium abundance, Jupiter interior models often invoke an outer stably stratified region due to helium rain. Additionally, Juno gravity data suggest a deeper, potentially stratified dilute core extending halfway through the planet. However, fits to Jupiter's gravitational data are non‐unique, and outstanding uncertainty over the equations of state for hydrogen and helium remain. Here, we use high‐resolution numerical magnetohydrodynamic simulations of Jupiter's magnetic field to place constraints on the extent of stable stratification within the planet. We find that compared to traditional interior models, an upper stably stratified layer between 0.9 and 0.95 Jupiter radii (RJ) helps to explain both Jupiter's dipolar magnetic field and zonal winds. In contrast, an extended dilute core that is entirely stably stratified (no convective layers) yields significantly worse fits to both. However, our models with extended deep stratification still generate dipolar magnetic fields if an upper stratified region is also present. Overall, we find that a planet with a dilute core i.e., strongly stably stratified is increasingly challenging to reconcile with Jupiter's magnetic field and winds. Thus if a dilute core is present, alternative modalities such as a fully convective dilute core, a complex multilayered interior structure, or double diffusive convection may be required.

The Sun’s internal structure and dynamics can be studied with helioseismology, which uses the Sun’s natural acoustic oscillations to build up a profile of the solar interior. We discuss how solar acoustic oscillations are affected by the Sun’s magnetic field. Careful observations of these effects can be inverted to determine the variations in the structure and dynamics of the Sun’s interior as the solar cycle progresses. Observed variations in the structure and dynamics can then be used to inform models of the solar dynamo, which are crucial to our understanding of how the Sun’s magnetic field is generated and maintained.

Constraining Europa's ice shell thickness with fundamental mode surface wave dispersion

A potential high-level radioactive waste (HLRW) disposal site in northwestern China was investigated to determine its suitability for such a use. The site is primarily covered with well-developed metamorphic granite rocks. The primary targets for geological repositories are three granite rock masses (I, II, and III). Only surface geological data were available from previous studies. The objective of this study was to evaluate the quality of the rock mass and identify any weak geological structures that could jeopardize this future underground repository. We used gravity and aeromagnetic data on a large scale to study the regional geological structures within and around the three rock masses. Subsequently, in 2009, a controlled source audio-frequency magnetotelluric (CSAMT) survey was conducted to study rock mass I in more detail. This paper introduces three-dimensional (3-D) tomography imaging of gravity and aeromagnetic data. 3-D tomography imaging was carried out on previously collected gravity and aeromagnetic data and, using the results of different depth slices, we evaluated the rock mass quality and interpreted the geology. The aeromagnetic depth slices show that at about 1-km deep in rock masses I and III there is a high magnetic susceptibility body, possibly caused by an early granite intrusion. The deep granite or metamorphic rocks appear as weak `magnetic anomalies in the magnetic slices. The results also indicate that rock mass II is more deeply buried and smaller than rock masses I and III. In the upper crust, inhomogeneous density variations have a strike direction consistent with that of the structure revealed in the aeromagnetic data. For rock mass I, we designed a CSAMT line to map small-scale outcropped faults to depth, and identify any hidden faults or weakened zones in the subsurface. Because of the highly resistive ground, we improved the electrode contact conditions by pouring salt water on both the transmitting and receiving electrodes. During data processing, we applied Hanning filtering to reduce static effects from lack of electrical homogeneity near the ground surface, and through experimentation found an optimum filter length to achieve maximum static effect reduction. Sounding data were inverted for geoelectrical cross sections using a two-dimensional smooth-model inversion method. Inversion artifacts were suppressed by imposing the model smoothness constraint. Several important results were revealed by the inversion: the inversion section correctly recognized two fractures at the surface, and also identified two new faults that were not observed during the previous geological survey. The inversion profile suggests that the narrow factures and deformation bands extend to a great depth. With further drilling confirmation, this interpretation will enhance understanding of the deep geological structures at the HLRW disposal site.

Helioseismology studies the structure and dynamics of the Sun's interior by observing oscillations on the surface. These studies provide information about the physical processes that control the evolution and magnetic activity of the Sun. In recent years, helioseismology has made substantial progress towards the understanding of the physics of solar oscillations and the physical processes inside the Sun, thanks to observational, theoretical and modeling efforts. In addition to the global seismology of the Sun based on measurements of global oscillation modes, a new field of local helioseismology, which studies oscillation travel times and local frequency shifts, has been developed. It is capable of providing 3D images of the subsurface structures and flows. The basic principles, recent advances and perspectives of global and local helioseismology are reviewed in this article.