Solar Internal Rotation Research Articles

Minuscule Corrections to Near-surface Solar Internal Rotation Using Mode Coupling

The observed solar oscillation spectrum is influenced by internal perturbations such as flows and structural asphericities. These features induce splitting of characteristic frequencies and distort the resonant-mode eigenfunctions. Global axisymmertric flow—differential rotation—is a very prominent perturbation. Tightly constrained rotation profiles as a function of latitude and radius are products of established helioseismic pipelines that use observed Dopplergrams to generate frequency-splitting measurements at high precision. However, the inference of rotation using frequency splittings does not consider the effect of mode coupling. This approximation worsens for modes with high angular degrees, as they become increasingly proximal in frequency. Since modes with high angular degrees probe the near-surface layers of the Sun, inversions considering coupled modes could potentially lead to more accurate estimates of rotation very close to the surface. In order to investigate if this is indeed the case, we perform inversions for solar differential rotation, considering coupling of modes for angular degrees 160 ≤ ℓ ≤ 300 in the surface gravity f-branch and first-overtone p modes. In keeping with the character of mode coupling, we carry out a nonlinear inversion using an eigenvalue solver. Differences in inverted profiles for frequency-splitting measurements from MDI and HMI are compared and discussed. We find that the corrections to the near-surface differential rotation profile, when accounting for mode-coupling effects, are smaller than 0.003 nHz and hence are insignificant. These minuscule corrections are found to be correlated with the solar cycle. We also present corrections to even-order splitting coefficients, which could consequently impact inversions for structure and magnetic fields.

The internal rotation of the Sun and its link to the solar Li and He surface abundances

The Sun serves as a natural reference for the modelling of the various physical processes at work in stellar interiors. Helioseismology results, which inform us on the characterization of the interior of the Sun (such as, for example, the helium abundance in its envelope), are, however, at odds with heavy element abundances. Moreover, the solar internal rotation and surface abundance of lithium have always been challenging to explain. We present results of solar models that account for transport of angular momentum and chemicals by both hydrodynamic and magnetic instabilities. We show that these transport processes reconcile the internal rotation of the Sun, its surface lithium abundance, and the helioseismic determination of the envelope helium abundance. We also show that the efficiency of the transport of chemicals required to account for the solar surface lithium abundance also predicts the correct value of helium, independently from a specific transport process.

Updated values of solar gravitational moments J2n using HMI helioseismic inference of internal rotation

ABSTRACT The solar gravitational moments J2n are important astronomical quantities whose precise determination is relevant for solar physics, gravitational theory and high precision astrometry, and celestial mechanics. Accordingly, we propose in the present work to calculate new values of J2n (for n = 1, 2, 3, 4, and 5) using recent two-dimensional rotation rates inferred from the high-resolution Solar Dynamics Observatory / Helioseismic and Magnetic Imager helioseismic data spanning the whole solar activity cycle 24. To this aim, a general integral equation relating J2n to the solar internal density and rotation is derived from the structure equations governing the equilibrium of slowly rotating stars. For comparison purpose, the calculations are also performed using rotation rates obtained from a recently improved analysis of Solar and Heliospheric Observatory / Michelson Doppler Imager heliseismic data for solar cycle 23. In agreement with earlier findings, the results confirmed the sensitivity of high-order moments (n > 1) to the radial and latitudinal distribution of rotation in the convective zone. The computed value of the quadrupole moment J2 (n = 1) is in accordance with recent measurements of the precession of Mercury’s perihelion deduced from high precision ranging data of the MESSENGER spacecraft. The theoretical estimate of the related solar oblateness Δ⊙ is consistent with the most accurate space-based determinations, particularly the one from Reuven Ramathy High-Energy Solar Spectroscopic Imager/Solar Aspect Sensor.

Lithium depletion and angular momentum transport in solar-type stars

Context. Transport processes occurring in the radiative interior of solar-type stars are evidenced by the surface variation of light elements, in particular 7Li, and the evolution of their rotation rates. For the Sun, inversions of helioseismic data indicate that the radial profile of angular velocity in its radiative zone is nearly uniform, which implies the existence of angular momentum transport mechanisms that are efficient over evolutionary timescales. While there are many independent transport models for angular momentum and chemical species, there is a lack of self-consistent theories that permit stellar evolution models to simultaneously match the present-day observations of solar lithium abundances and radial rotation profiles. Aims. We explore how additional transport processes can improve the agreement between evolutionary models of rotating stars and observations for 7Li depletion, the rotation evolution of solar-type stars, and the solar rotation profile. Methods. Models of solar-type stars are computed including atomic diffusion and rotation-induced mixing with the code STAREVOL. We explore different additional transport processes for chemicals and for angular momentum such as penetrative convection, tachocline mixing, and additional turbulence. We constrain the resulting models by simultaneously using the evolution of the surface rotation rate and 7Li abundance in the solar-type stars of open clusters with different ages, and the solar surface and internal rotation profile as inverted from helioseismology when our models reach the age of the Sun. Results. We show the relevance of penetrative convection for the depletion of 7Li in pre-main sequence and early main sequence stars. The rotational dependence of the depth of penetrative convection yields an anti-correlation between the initial rotation rate and 7Li depletion in our models of solar-type stars that is in agreement with the observed trend. Simultaneously, the addition of an ad hoc vertical viscosity νadd leads to efficient transport of angular momentum between the core and the envelope during the main sequence evolution and to solar-type models that match the observed profile of the Sun. We also self-consistently compute for the first time the thickness of the tachocline and find that it is compatible with helioseismic estimations at the age of the Sun, but we highlight that the associated turbulence does not allow the observed 7Li depletion to be reproduced. The main sequence depletion of 7Li in solar-type stars is only reproduced when adding a parametric turbulent mixing below the convective envelope. Conclusions. The need for additional transport processes in stellar evolution models for both chemicals and angular momentum in addition to atomic diffusion, meridional circulation, and turbulent shear is confirmed. We identify the rotational dependence of the penetrative convection as a key process. Two additional and distinct parametric turbulent mixing processes (one for angular momentum and one for chemicals) are required to simultaneously explain the observed surface 7Li depletion and the solar internal rotation profile. We highlight the need of additional constraints for the internal rotation of young solar-type stars and also for the beryllium abundances of open clusters in order to test our predictions.

Inference of Solar Rotation from Perturbations of Acoustic Mode Eigenfunctions

Abstract Today’s picture of the internal solar rotation rate profile results essentially from helioseismic analyses of frequency splittings of resonant acoustic waves. Here we present another, complementary estimation of the internal solar rotation rate using the perturbation of the shape of the acoustic waves. For this purpose, we extend a global helioseismic approach developed previously for the investigation of the meridional flow to work on the components of the differential rotation. We discuss the effect of rotation on mode eigenfunctions and the observables based thereon. Based on a numerical study using a simulated rotation rate profile, we tailor an inversion approach and also consider the case of the presence of an additional meridional flow. This inversion approach is then applied to data from the Michelson Doppler Imager (MDI) on board the Solar Heliospheric Observatory and the Helioseismic and Magnetic Imager (HMI) on board the Solar Dynamics Observatory. In the end, rotation rate profiles estimated from eigenfunction perturbation and frequency splittings are compared. The rotation rate profiles from the two different approaches are qualitatively in good agreement, especially for the MDI data. Significant differences are obtained at high latitudes >50° and near the subsurface. The result from HMI data shows larger discrepancies between the different methods. We find that the two global helioseismic approaches provide complementary methods for measuring the solar rotation. Comparing the results from different methods may help to reveal systematic influences that affect analyses based on eigenfunction perturbations, like meridional flow measurements.

Analysis of beta-decay data acquired at the Physikalisch-Technische Bundesanstalt: Evidence of a solar influence

According to an article entitled Disproof of solar influence on the decay rates of 90Sr/90Y by Kossert and Nähle of the Physikalisch-Technische Bundesanstalt (PTB) [1], the PTB measurements show no evidence of variability. We show that, on the contrary, those measurements reveal strong evidence of variability, including an oscillation at 11 year−1 that is suggestive of an influence of internal solar rotation. An analysis of radon beta-decay data acquired at the Geological Survey of Israel (GSI) Laboratory for the same time interval yields strong confirmation of this oscillation.

Mode Frequencies from 17, 15 and 2 Years of GONG, MDI, and HMI Data

We present results from fitting all the available data to date from the three major helioseismic instruments: MDI, GONG and HMI. These data were fitted using an innovative and independent methodology devised a few years ago. The mode fitting was carried out on time series of varying lengths (1×, 2×, 4×, 8×, 16×, 32×, 64 × 72 day-long), using co-eval epochs for all three data sets. By fitting time series of varying lengths, one trades off some temporal resolution for a better precision. We present a comparison of these results, and discuss the potential sources of the residual small discrepancies. We also present inferences from these results on the determination of the solar internal rotation and changes with epoch and thus activity levels.

The rotation rate and its evolution derived from improved mode fitting and inversion methodology

We present inferences of the internal solar rotation rate and its evolution during solar cycle 23. A full solar cycle of MDI observations have been analyzed using an improved fitting methodology and using time series of various lengths, up to a single 4,608 day long epoch (64 times 72 days or 12.6 yr). We used time series of spherical harmonic coefficients computed by the MDI group, including those resulting from using their improved spatial decomposition. This decomposition includes our best estimate of the image plate scale and of the MDI instrumental image distortion. The leakage matrix used in the fitting includes the effect of the distortion of the eigenfunctions by the solar differential rotation, while the undistorted leakage matrix was itself carefully reviewed and independently recomputed. Rotation inversions were carried out for all available mode sets, fitted for that epoch, including the MDI and GONG "pipe-line" values. The improved inversion method uses an iterative methodology based on a least-squares regularization, but with an optimal model grid determined by the actual information in the input set. This method also allows us to use an optimized irregular grid, with a variable number of latitudes at different depths.

SEISMIC AND DYNAMICAL SOLAR MODELS. I. THE IMPACT OF THE SOLAR ROTATION HISTORY ON NEUTRINOS AND SEISMIC INDICATORS

Solar activity and helioseismology show the limitation of the standard solar model and call for the inclusion of dynamical processes in both convective and radiative zones. We concentrate here on the radiative zone and first show the sensitivity of boron neutrinos to the microscopic physics included in solar models. We confront the neutrino predictions of the seismic model to all the detected neutrino fluxes. Then we compute new models of the Sun including a detailed transport of angular momentum and chemicals due to internal rotation that includes meridional circulation and shear induced turbulence. We use two stellar evolution codes: CESAM and STAREVOL to estimate the different terms. We follow three temporal evolutions of the internal rotation differing by their initial conditions: very slow, moderate and fast rotation, with magnetic braking at the arrival on the main sequence for the last two. We find that the meridional velocity in the present solar radiative zone is extremely small in comparison with those of the convective zone, smaller than 10^-6 cm/s instead of m/s. All models lead to a radial differential rotation profile but with a significantly different contrast. We compare these profiles to the presumed solar internal rotation and show that if meridional circulation and shear turbulence were the only mechanisms transporting angular momentum within the Sun, a rather slow rotation in the young Sun is favored. The transport by rotation slightly influence the sound speed profile but its potential impact on the chemicals in the transition region between radiation and convective zones. This work pushes us to pursue the inclusion of the other dynamical processes to better reproduce the present observable and to describe the young active Sun. We also need to get a better knowledge of solar gravity mode splittings to use their constraints.

The Sun as a fundamental calibrator of stellar evolution

AbstractThe Sun is unique amongst stars in having a precisely determined age which does not depend on the modelling of stellar evolution. Furthermore, other global properties of the Sun are known to much higher accuracy than for any other star. Also, helioseismology has provided detailed determination of the solar internal structure and rotation. As a result, the Sun plays a central role in the development and test of stellar modelling. Here I discuss solar modelling and its application to tests of asteroseismic techniques for stellar age determination.

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 ℓ≤3. In the work presented here, we used an 11.5-year-long time series to compute the rotational frequency splittings for modes ℓ≤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.

What Prevents Internal Gravity Waves from Disturbing the Solar Uniform Rotation?

Internal gravity waves (IGWs) are naturally produced by convection in stellar envelopes, and they could be an important mechanism for transporting angular momentum in the radiative interiors of stars. Prior work has established that they could operate over a short enough timescale to explain the internal solar rotation as a function of depth. We demonstrate that the natural action of IGWs is to produce large-scale oscillations in the solar rotation as a function of depth, which is in marked contrast to the nearly uniform rotation in the outer radiative envelope of the Sun. An additional angular momentum transport mechanism is therefore required, and neither molecular nor shear-induced turbulent viscosity is sufficient to smooth out the profile. Magnetic processes, such as the Tayler-Spruit dynamo, could flatten the rotation profile. We therefore conclude that IGWs must operate in conjunction with magnetic angular momentum transport processes if they operate at all. Furthermore, both classes of mechanisms must be inhibited to some degree by mean molecular weight gradients in order to explain the recent evidence for a rapidly rotating embedded core in the Sun, should it be confirmed by a further analysis of solar g-mode oscillations.

Solar p modes of high degree l: coupling by differential rotation

Accurate measurements of solar p-mode frequencies and frequency splittings at high degree l require an adequate theoretical knowledge of the effects of mode coupling, induced by the variation with latitude of the angular velocity of the solar internal rotation. Earlier results for expansion coefficients of composite solutions (coupling coefficients) are due to Woodard. In this paper, the analysis is extended to allow for the dependence of the differential rotation on depth, and the result is expressed in terms of measurable quantities (the rotational splitting coefficients), which makes it convenient for diagnostic purposes. The analysis is based on the approach of quasi-degenerate perturbation theory, and is extended further to address possible effects of mode coupling in the observational line profiles. It is shown, using approximations applicable at high degree l, that the expected line profiles of composite modes in the observational power spectra are not distorted by mode coupling.

A dynamo model for axisymmetric and non-axisymmetric solar magnetic fields

Increasing observations are becoming available about a relatively weak, but persistent, non-axisymmetric magnetic field co-existing with the dominant axisymmetric field on the Sun. It indicates that the non-axisymmetric magnetic field plays an important role in the origin of solar activity. A linear non-axisymmetric alpha2-Omega dynamo model is set up to discuss the characteristics of the axisymmetric m=0 and the first non-axisymmetric m=1 modes and to provide further the theoretical bases to explain the active longitude, flip-flop and other non-axisymmetric phenomena. The model consists of a updated solar internal differential rotation, a turbulent diffusivity varied with depth and an alpha-effect working at the tachocline in rotating spherical systems. The difference between the alpha2-Omega and the alpha-Omega models and the conditions to favor the non-axisymmetric modes with the solar-like parameters are also presented.

The Alpha Effect and the Observed Twist and Current Helicity of Solar Magnetic Fields

We present a straightforward comparison of model calculations for the α-effect, helicities, and magnetic field line twist in the solar convection zone with magnetic field observations at atmospheric levels. The model calculations are carried out in a mixing-length approximation for the turbulence with a profile of the solar internal rotation rate obtained from helioseismic inversions. The magnetic field data consist of photospheric vector magnetograms of 422 active regions for which spatially-averaged values of the force-free twist parameter and of the current helicity density are calculated, which are then used to determine latitudinal profiles of these quantities. The comparison of the model calculations with the observations suggests that the observed twist and helicity are generated in the bulk of the convection zone, rather than in a layer close to the bottom. This supports two-layer dynamo models where the large-scale toroidal field is generated by differential rotation in a thin layer at the bottom while the α-effect is operating in the bulk of the convection zone. Our previous observational finding was that the moduli of the twist factor and of the current helicity density increase rather steeply from zero at the equator towards higher latitudes and attain a certain saturation at about 12 – 15∘. In our dynamo model with algebraic nonlinearity, the increase continues, however, to higher latitudes and is more gradual. This could be due to the neglect of the coupling between small-scale and large-scale current and magnetic helicities and of the latitudinal drift of the activity belts in the model.

A Flux Tube Solar Dynamo Model Based on the Competing Role of Buoyancy and Downflows

A magnetic flux tube can be considered both as a separate body and as a confined field. As a field, it is affected by both differential rotation (Ω-effect) and cyclonic convection (α-effect). As a body, the tube experiences not only a buoyant force, but also a dynamic pressure due to downflows above the tube. These two competing dynamic effects are incorporated into the α-Ω dynamo equations through the total magnetic turbulent diffusivity, leading to a flux tube dynamo operating in the convection zone. We analyze and solve the extended dynamo equations in the linear approximation by adopting the observed solar internal rotation and assuming a downflow effect derived from numerical simulations of a solar convection zone. The model reproduces the 22 yr cycle period; the extended butterfly diagram with the confinement of strong activity to low heliographic latitudes |Φ| ≤ 35°; the evidence that at low latitudes the radial field is in an approximately π phase lag compared to the toroidal field at the same latitude; the evidence that the poleward branch is in a π/2 phase lag with respect to the equatorward branch; and the evidence that most of the magnetic flux is present in an intermittent form, concentrated into strong flux tubes.

Dynamical Processes Induced by the Internal Solar Rotation

Helioseismology has been developed to extract internal sound speed, rotation and magnetic field. Such informations will put strong constraints on the present missing dynamical phenomena. Starting from the present observations on the rotation, I discuss quantitatively two important regions, the transition region between radiation and convection, called tachocline and the radiative region. In the tachocline, lithium indicator shows that a turbulent mixing is certainly present in main sequence solar like stars. on the other end, a detailed study of pre-main sequence stars encourages also the introduction of magnetic effects in addition to such a term below 10 Myr. For the solar radiative zone, the difficulty to determine the rotation profile is first noticed and discussed showing the great advantage of the gravity modes or mixed modes in addition to the acoustic modes. A possible profile, decreasing from 0.4 to 0.2 R⊙ and then increasing in the very central region is suggested by the present observations. The consequences in terms of dynamical processes are discussed.

The Internal Solar Rotation from Helioseismology

Helioseismology has provided detailed inferences about the internal rotation of the Sun, and hence stringent constraints on any attempt to understand the properties and evolution of stellar rotation. Here I briefly discuss the techniques used in the analysis and review the results that have been obtained. Strikingly, these results are markedly different from the predictions made before the helioseismic data became available, emphasizing the difficulties in the modelling of phenomena as complex as stellar internal rotation.

Gravity Wave–driven Flows in the Solar Tachocline. II. Stationary Flows

The effects of gravity waves on the mean radial differential rotation profile in the solar tachocline are studied, including the effect of a uniform, toroidal magnetic field. Vertical transport of horizontal momentum arises from the radiative damping of inwardly traveling waves that are generated by low-frequency, convective fluid motions. By considering two-wave and one-wave interactions, the radiatively damped gravity waves are shown to accentuate the shear in the mean radial differential rotation. In the presence of a strong horizontal magnetic field, internal gravity waves become nearly Alfvenic and cannot propagate downward through the tachocline. For a magnetic field that is weak enough to permit wave propagation, the mean shear profile is shown to be smoother than that obtained in the case of purely hydrodynamic waves. The implications of our results for gravity-wave forcing of the internal solar rotation are discussed.

Helicity and the solar dynamo

The central mechanism in traditional mean-field dynamo theory is the α-effect, and it has been found that the presence of kinetic or magnetic helicities is favourable for the effect, which corresponds to the simultaneous generation of magnetic helicities in the mean field and in the fluctuations, the generation rates being equal in magnitude and opposite in sign. Generally, the two helicities generated by the α-effect, that in the mean field and that in the fluctuations, have either to be dissipated in the generation region or to be transported out of this region. The latter presumably leads to the observed appearance of magnetic helicity in the solar atmosphere, which thus provides valuable information on dynamo processes inaccessible to in situ measurements. We have included details of two numerical dynamo studies in the present review, one for a “laminar” dynamo, where no averaging is applied, the other for a mean-field dynamo. In the first case the full nonlinear system of the incompressible MHD equations is studied in idealized rectangular geometry, with an external forcing of the Roberts type driving a flow in the form of an array of convection-like rolls. Defining mean fields by appropriate averages, it is found that there is a segregation of magnetic helicity between the mean field and the fluctuations similar to that predicted by the mean-field theory of the α-effect. The mean-field calculations are done in a quasi-linear approximation for the turbulence, for realistic spherical geometry, with compressibihy included and using a profile of the solar internal rotation rate obtained from helioseismic inversions. The results are compared with observations, concentrating on the observational finding that the moduli of the averaged values of the force-free twist parameter α ff and and the current helicity H C increase from zero at the equator towards higher latitudes and attain a certain saturation level at middle latitudes (at about 20°–30°). On the assumption that the α-effect is operating in a thin spherical shell, the best coincidence between calculated and observed quantities is found for α-effect operation close to the bottom of the convection zone.