91 pages, 32 figures: Accepted by Living Reviews in Solar Physics 10th Feb 2009 Published version http://www.livingreviews.org/lrsp-2009-1 (Updated to reflect publication info and improve fig 27)
This article surveys the development of observational understanding of the interior rotation of the Sun and its temporal variation over approximately forty years, starting with the 1960s attempts to determine the solar core rotation from oblateness and proceeding through the development of helioseismology to the detailed modern picture of the internal rotation deduced from continuous helioseismic observations during solar cycle 23. After introducing some basic helioseismic concepts, it covers, in turn, the rotation of the core and radiative interior, the "tachocline" shear layer at the base of the convection zone, the differential rotation in the convection zone, the near-surface shear, the pattern of migrating zonal flows known as the torsional oscillation, and the possible temporal variations at the bottom of the convection zone. For each area, the article also briefly explores the relationship between observations and models.
We describe a simple method to study the dependence of the solar properties on a generic (small) modification the physical inputs adopted in standard solar models calculations.
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…
A major outstanding problem in solar physics is the confinement of the solar tachocline, the thin shear layer that separates nearly solid-body rotation in the radiative interior from strong differential rotation in the convection zone. Here, we present the first 3-D, global solar simulation in which a tachocline is confined by a self-excited dynamo. The non-axisymmetric magnetism is initially built in the convection zone and then diffusively imprints downward. Additionally, the field is locally amplified throughout the radiative interior by vigorous horizontal motions that arise from equatoria…
We compute the change in the Lorentz force integrated over the outer solar atmosphere implied by observed changes in vector magnetograms that occur during large, eruptive solar flares. This force perturbation should be balanced by an equal and opposite force perturbation acting on the solar photosphere and solar interior. The resulting expression for the estimated force change in the solar interior generalizes the earlier expression presented by Hudson, Fisher and Welsch (CS-383, ASP, 221, 2008), providing horizontal as well as vertical force components, and provides a more accurate result for…
The solar gravity modes are the best probes to improve our knowledge on the solar interior, as they spend most of their time in the radiative zone, which represents 98% of the solar mass. Many attempts have been led to observe them using different techniques: either individually, then adding some statistical approach or more recently, globally leading to the detection of the signature of asymptotical properties of these modes. Then, several theoretical works have been done to quantify the effect of detecting g-mode on solar modeling and on the rotation profile. We will give here an update on t…
Acoustic waves generated by a point source in stratified plasma are considered in this paper. Analytical parametric solution for monochromatic source is derived for plane-parallel polytrope model of the solar interior. The solution is used to gain insight into the properties of the generated wavefront as a function of excitation frequency and depth. A slowly varying pressure perturbation moving in upper layers of solar photosphere with supersonic speed is also considered. It is shown to excite acoustic waves putting certain restrictions upon their geometry of the generated wavefront. The resul…
We review some major open issues in the current modelling of low and intermediate mass, main sequence stars based on seismological studies. The solar case was discussed in a companion paper, here several issues specific to other stars than the Sun are illustrated with a few stars observed with CoRoT and expectations from Kepler data.
Helioseismology has taught us a great deal about the stratification and kinematics of the solar interior, sufficient for us to embark upon dynamical studies more detailed than have been possible before. The most sophisticated studies to date have been the very impressive numerical simulations of the convection zone, from which, especially in recent years, a great deal has been learnt. Those simulations, and the seismological evidence with which they are being confronted, are reviewed elsewhere in this volume. Our understanding of the global dynamics of the radiative interior of the Sun is in a…
The high performance photometric data obtained with space mission CoRoT offer the opportunity to efficiently constrain our models for the stellar interior of solar-like pulsating stars. On the occasion of the analysis of the oscillations of solar-like pulsator HD 49385, a G0-type star in an advanced stage of evolution, we revisit the phenomenon of the avoided crossings. Christensen-Dalsgaard proposed a simple analogy to describe an avoided crossing between two modes. We here present an extension of this analogy to the case of $n$ modes, and show that it should lead, in certain cases, to a char…
One of the largest uncertainties in stellar structure and evolution theory is the transport of angular momentum in the stellar interiors. Asteroseismology offers a powerful tool for measuring the internal rotation frequencies of pulsating stars, but the number of such measurements has remained few for $\gtrsim 3\,{\rm M}_\odot$ main-sequence stars. In this work, we compile a list of 52 slowly pulsating B stars for which the interior rotation has been measured asteroseismically. The measurements of the spin parameters, which describe the relative importance of rotation, for the gravito-inertial…
We provide a status report on the determination of stellar ages from asteroseismology for stars of various masses and evolutionary stages. The ability to deduce the ages of stars with a relative precision of typically 10 to 20% is a unique opportunity for stellar evolution and also of great value for both galactic and exoplanet studies. Further, a major uncalibrated ingredient that makes stellar evolution models uncertain, is the stellar interior rotation frequency $Ω(r)$ and its evolution during stellar life. We summarize the recent achievements in the derivation of $Ω(r)$ for different types…
We review some major, open issues in the current modelling of low and intermediate mass, main sequence stars based on seismological studies. In the present paper, the solar case is discussed together with current problems that are common to the Sun and stars with a structure similar to that of the Sun. Several additional issues specific to main sequence stars other than the Sun are reviewed and illustrated with a few stars observed with CoRoT in a companion paper.
The Sun supports a rich spectrum of internal waves that are continuously excited by turbulent convection. The GONG network and the MDI/SOHO space instrument provide an exceptional data base of spatially-resolved observations of solar oscillations, covering an entire sunspot cycle (11 years). Local helioseismology is a set of tools for probing the solar interior in three dimensions using measurements of wave travel times and local mode frequencies. Local helioseismology has discovered (i) near-surface vector flows associated with convection (ii) 250 m/s subsurface horizontal outflows around sun…
We report on a systematic center-to-limb variation in measured helioseismic travel times, which must be taken into account for an accurate determination of solar interior meridional flows. The systematic variation, found in time-distance helioseismology analysis using SDO/HMI and SDO/AIA observations, is different in both travel-time magnitude and variation trend for different observables. It is not clear what causes this systematic effect. Subtracting the longitude-dependent east-west travel times, obtained along the equatorial area, from the latitude-dependent north-south travel times, obtai…
To date, no self-consistent numerical simulation of the solar interior has succeeded in reproducing the observed thinness of the solar tachocline, and the persistence of uniform rotation beneath it. Although it is known that the uniform rotation can be explained by the presence of a global-scale confined magnetic field, numerical simulations have thus far failed to produce any solution where such a field remains confined against outward diffusion. We argue that the problem lies in the choice of parameters for which these numerical simulations have been performed. We construct a simple analytic…
With the assumption of local Tsallis equilibrium, the newly defined gravitational temperature is calculated in the solar interior, whose distribution curve can be divided into three parts, the solar core region, radiation region and convection region, in excellent agreement with the solar hierarchical structure. By generalizing the Fourier law, one new mechanism of heat conduction, based on the gradient of the gravitational temperature, is introduced into the astrophysical system. This mechanism is related to the self-gravity of such self-gravitating system whose characteristic scale is large …
Context: The internal rotation of the Sun constitutes a fundamental constraint when modelling angular momentum transport in stellar interiors. In addition to the more external regions of the solar radiative zone probed by pressure modes, measurements of rotational splittings of gravity modes would offer an invaluable constraint on the rotation of the solar core. Aims: We study the constraints that a measurement of the core rotation rate of the Sun could bring on magnetic angular momentum transport in stellar radiative zones. Results: We first show that models computed with angular momentum tra…
We observe the abrupt end of solar activity cycles at the Sun's equator by combining almost 140 years of observations from ground and space. These "terminator" events appear to be very closely related to the onset of magnetic activity belonging to the next sunspot cycle at mid-latitudes and the polar-reversal process at high-latitudes. Using multi-scale tracers of solar activity we examine the timing of these events in relation to the excitation of new activity and find that the time taken for the solar plasma to communicate this transition is of the order of one solar rotation, but could be s…
Almost 100 years ago Sir Arthur Eddington noted that the interiors of stars were inaccessible to observations. The advent of helio- and asteroseismology has completely changed this assessment. Helioseismology has provided very detailed information about the interior structure and dynamics of the Sun, highlighting remaining issues in our understanding of the solar interior. In the last decade extensive observations of stellar oscillations, in particular from space photometry, have provided very detailed information about the global and internal properties of stars. Here I provide an overview of…
Solar subsurface zonal and meridional-flow profiles during the rising phase of solar cycle 24 are studied using time-distance helioseismology technique. The faster zonal bands in the torsional-oscillation pattern show strong hemispheric asymmetries and temporal variations in both width and speed. The faster band in the northern hemisphere is located closer to the equator than the band in the southern hemisphere, and migrates past the equator when the magnetic activity in the southern hemisphere is reaching maximum. The meridional-flow speed decreases substantially with the increase of magnetic…
The possibility of measuring the internal rotation of the Sun and stars thanks to helio- and asteroseismology offers tremendous constraints on hydro- and magnetohydrodynamical processes acting in stellar interiors. Understanding the processes responsible for the transport of angular momentum in stellar interiors is crucial as they will also influence the transport of chemicals and thus the evolution of stars. Here we present some of the key results obtained in both fields and how detailed seismic analyses can provide stringent constraints on the physics of angular momentum transport in the int…
Helioseismology, the study of global solar oscillations, has proved to be an extremely powerful tool for the investigation of the internal structure and dynamics of the Sun. Studies of time changes in frequency observations of solar oscillations from helioseismology experiments on Earth and in space have shown, for example, that the Sun's shape varies over solar cycle timescales. In particular, far-reaching inferences about the Sun have been obtained by applying inversion techniques to observations of frequencies of oscillations. The results, so far, have shown that the solar structure is rema…
Context: Recent asteroseismic observations have led to the determination of rotational frequency splittings for l=1 mixed modes in red giants. Aims: We investigate how these observed splittings can constrain the modelling of the physical processes transporting angular momentum in stellar interiors. Methods: We first compare models including a comprehensive treatment of shellular rotation only, with the rotational splittings observed for the red giant KIC 8366239. We then study how these asteroseismic constraints can give us information about the efficiency of an additional mechanism for the in…
Motivated by the interface model for the solar dynamo, this paper explores the complex magnetohydrodynamic interactions between convective flows and shear-driven instabilities. Initially, we consider the dynamics of a forced shear flow across a convectively-stable polytropic layer, in the presence of a vertical magnetic field. When the imposed magnetic field is weak, the dynamics are dominated by a shear flow (Kelvin-Helmholtz type) instability. For stronger fields, a magnetic buoyancy instability is preferred. If this stably stratified shear layer lies below a convectively unstable region, th…
We investigated the solar cycle dependency on the presence and periodicity of the Quasi-Biennial Oscillation (QBO). Using helioseismic techniques, we used solar oscillation frequencies from the Global Oscillations Network Group (GONG), Michelson Doppler Imager (MDI) and Helioseismic & Magnetic Imager (HMI) in the intermediate-degree range to investigate the frequency shifts over Cycles 23 and 24. We also examined two solar activity proxies, the F10.7 index and the MgII index, for the last four solar cycles to study the associated QBO. The analyses were performed using Empirical Mode Decomposit…
We constrain the velocity spectral distribution of global-scale solar convective cells at depth using techniques of local helioseismology. We calibrate the sensitivity of helioseismic waves to large-scale convective cells in the interior by analyzing simulations of waves propagating through a velocity snapshot of global solar convection via methods of time-distance helioseismology. Applying identical analysis techniques to observations of the Sun, we are able to bound from above the magnitudes of solar convective cells as a function of spatial convective scale. We find that convection at a dep…
The equation of state (EOS) of the solar interior is accurately and smoothly determined from \textit{ab initio} simulations named quantum Langevin molecular dynamics (QLMD) in the pressure range of $58 \leq P \leq 4.6\times10^5$ Mbar at the temperature range of $1 \leq T \leq 1500$ eV. The central pressure is calculated, and compared with other models. The effect of heavy elements such as carbon and oxygen on the EOS is also discussed.
In this article, we report an evidence of very high and statistically significant relationship between hemispheric asymmetry in solar coronal rotation rate and solar activity. Our approach is based on cross correlation of hemispheric asymmetry index (AI) in rotation rate with annual solar activity indicators. To obtain hemispheric asymmetry in solar rotation rate, we use solar full disc (SFD) images at 30.4 nm, 19.5 nm, and 28.4 nm wavelengths for 24th Solar Cycle i.e., for the period from 2008 to 2018, as recorded by the Solar Terrestrial Relations Observatory (STEREO) space mission. Our anal…
Effects of the radiative interior on solar inertial modes
Max-Planck-Institut für Sonnensystemforschung（马克斯·普朗克太阳系统研究所）；Institut für Astrophysik und Geophysik, Georg-August-Universität Göttingen（哥廷根大学天体物理与地球物理研究所）
Context. Solar eruptions and high flare activity often accompany the rapid rotation of sunspots. The study of sunspot rotation and the mechanisms driving this motion are therefore key to our understanding of how the solar atmosphere attains the conditions necessary for large energy release. Aims. We aim to demonstrate and investigate the rotation of sunspots in a 3D numerical experiment of the emergence of a magnetic flux tube as it rises through the solar interior and emerges into the atmosphere. Furthermore, we seek to show that the sub-photospheric twist stored in the interior is injected i…
A comprehensive understanding of the solar magnetic cycle requires detailed modeling of the solar interior including the maintenance and variation of large scale flows (differential rotation and meridional flow), the solar dynamo and the flux emergence process connecting the magnetic field in the solar convection zone with magnetic field in the photosphere and above. Due to the vast range of time and length scales encountered, a single model of the entire convection zone is still out of reach. However, a variety of aspects can be modeled through a combined approach of 3D MHD models and simplif…
Until the last few decades, investigations of stellar interiors had been restricted to theoretical studies only constrained by observations of their global properties and external characteristics. However, in the last thirty years the field has been revolutionized by the ability to perform seismic investigations of stellar interiors. This revolution begun with the Sun, where helioseismology has been yielding information competing with what can be inferred about the Earth's interior from geoseismology. The last two decades have witnessed the advent of asteroseismology of solar-like stars, thank…
Stellar differential rotation can be separated into two main regimes: solar-like when the equator rotates faster than the poles and anti-solar when the polar regions rotate faster than the equator. We investigate the transition between these two regimes with 3-D numerical simulations of rotating spherical shells. We conduct a systematic parameter study which also includes models from different research groups. We find that the direction of the differential rotation is governed by the contribution of the Coriolis force in the force balance, independently of the model setup (presence of a magnet…
Is the Sun likely to have a more opaque interior than previously thought? The solar oxygen (or abundance) problem can be solved with higher interior opacities, reconciling abundance analyses based on 3D convective atmospheres with the helioseismic structure of the solar interior. This has been known for more than a decade, but last year we learned that the absorption by just iron may contribute 7\% more to the solar opacity at the bottom of the convection zone than predicted by any opacity calculation so far, and by OP05 in particular. I find that artificial changes to the absorption (calibrat…
The latitudinal distributions of the yearly mean rotation rates measured respectively by Suzuki in 1998 and 2012 and Pulkkinen $\&$ Tuominen in 1998 are utilized to investigate internal-cycle variation of solar differential rotation. The rotation rate at the solar Equator seems to decrease since cycle 10 onwards. The coefficient $B$ of solar differential rotation, which represents the latitudinal gradient of rotation, is found smaller in the several years after the minimum of a solar cycle than in the several years after the maximum time of the cycle, and it peaks several years after the maxim…
We obtain lower limits on the amplitude of convective velocities in the deep solar convection zone based only on the observed properties of the differential rotation and meridional circulation together with simple and robust dynamical balances obtained from the fundamental MHD equations. The linchpin of the approach is the concept of gyroscopic pumping whereby the meridional circulation across isosurfaces of specific angular momentum is linked to the angular momentum transport by the convective Reynolds stress. We find that the amplitude of the convective velocity must be at least 30 m s$^{-1}…
Solar activity undergoes a variation over time scales of several months known as Rieger-type periodicity, which usually occurs near maxima of sunspot cycles. An early analysis showed that the periodicity appears only in some cycles, and is absent in other cycles. But the appearance/absence during different cycles has not been explained. We performed a wavelet analysis of sunspot data from the Greenwich Royal Observatory and the Royal Observatory of Belgium during cycles 14-24. We found that the Rieger-type periods occur in all cycles, but they are cycle-dependent: shorter periods occur during …
Context. The relation between solar surface rotation and sunspot activity still remains open. Sunspot activity has dramatically reduced in solar cycle 24 and several solar activity indices and flux measurements experienced unprecedentedly low levels during the last solar minimum. Aims. We aim to reveal the momentary variation of solar surface rotation, especially during the recent years of reducing solar activity. Methods. We used a dynamic, differentially rotating reference system to determine the best-fit annual values of the differential rotation parameters of active longitudes of solar X-r…
Convection in the solar interior is thought to comprise structures on a spectrum of scales. This conclusion emerges from phenomenological studies and numerical simulations, though neither covers the proper range of dynamical parameters of solar convection. Here, we analyze observations of the wavefield in the solar photosphere using techniques of time-distance helioseismology to image flows in the solar interior. We downsample and synthesize 900 billion wavefield observations to produce 3 billion cross-correlations, which we average and fit, measuring 5 million wave travel times. Using these t…
Rotational shear in Sun-like stars is thought to be an important ingredient in models of stellar dynamos. Thanks to helioseismology, rotation in the Sun is characterized well, but the interior rotation profiles of other Sun-like stars are not so well constrained. Until recently, measurements of rotation in Sun-like stars have focused on the mean rotation, but little progress has been made on measuring or even placing limits on differential rotation. Using asteroseismic measurements of rotation we aim to constrain the radial shear in five Sun-like stars observed by the NASA Kepler mission: KIC0…
We use helioseismic data from ground and space-based instruments to analyze how solar rotation has changed since the beginning of solar Cycle 23 with emphasis on studying the differences between Cycles 23 and 24. We find that the nature of solar rotation is indeed different for the two cycles. While the changes in the latitudinally independent component follows solar-cycle indices, some of the other components have a more complicated behavior. There is a substantial change in the behavior of the solar zonal flows and their spatial gradients too. While the zonal flows are in general weaker in C…
(abridged) Context: Late-type stars such as the Sun rotate differentially due to the interaction of turbulent convection and rotation. Aims: The aim of the study is to investigate the effects of the thermal Prandtl number on the transition from anti-solar (slow equator, fast poles) to solar-like (fast equator, slow poles) differential rotation. Methods: Three-dimensional hydrodynamic and magnetohydrodynamic simulations in semi-global spherical wedge geometry are used to model convection zones of solar-like stars. Results: The overall convective velocity amplitude increases as the Prandtl numbe…
Solar-cycle related variation of differential rotation is investigated through analyzing the rotation rates of magnetic fields, distributed along latitudes and varying with time at the time interval of August 1976 to April 2008. More pronounced differentiation of rotation rates is found to appear at the ascending part of a Schwabe cycle than at the descending part on an average. The coefficient $B$ in the standard form of differential rotation, which represents the latitudinal gradient of rotation, may be divided into three parts within a Schwabe cycle. Part one spans from the start to the $4^…
In recent years, a revised set of solar abundances has led to a discrepancy in the sound-speed profile between helioseismology and theoretical solar models. Conventional solutions require additional mechanisms for energy transport within the Sun. Vincent et al. have recently suggested that dark matter with a momentum or velocity dependent cross section could provide a solution. In this work, we consider three models of dark matter with such cross sections and their effect on the stellar structure. In particular, the three models incorporate dark matter particles interacting through an electrom…
Helioseismology shows that the regions occupied by convection and differential rotation inside the sun almost coincide. This supports the leading theoretical concept for the origin of differential rotation as a result of interaction between convection and rotation. This talk outlines the current state of the differential rotation theory. Numerical models based on the theory reproduce the observed solar rotation quite closely. The models also compute meridional flow and predict that the flow at the bottom of the convection zone is not small compared to the surface. Theoretical predictions for s…
Variations of frequency separations of low-degree p-modes are studied over the solar activity cycle. The separations studied are obtained from the frequencies of low-degree p-modes of the Global Oscillation Network Group (GONG). 10.7 cm radio flux is used as an index of solar activity. Small separations of the p-mode frequencies are considered to be mainly dependent on the conditions in stellar interiors. Thus they could be applied to diagnose the changes in the stellar interior. Our calculation results show that the magnitudes of variations of the mean large separations are less than 1 $σ$ ov…
The solar photosphere, chromosphere and corona are known to rotate differentially as a function of latitude. To date, it is unclear if the solar transition region also rotates differentially. In this paper, we investigate differential rotational profile of solar transition region as a function of latitude, using solar full disk (SFD) images at 30.4 nm wavelength recorded by Extreme Ultraviolet Imager (EUVI) onboard Solar Terrestrial Relations Observatory (STEREO) space mission for the period from 2008 to 2018 (Solar Cycle 24). Our investigations show that solar transition region rotates differ…
A Comprehensive Simulation of Solar Wind Formation from the Solar Interior: Significant Cross-field Energy Transport by Interchange Reconnection near the Sun
Long-term modulation of solar differential rotation was studied with data from Mt. Wilson and our original observations during Solar Cycles 16 through 23. The results are: i) The global B-value (i.e. latitudinal gradient of differential rotation), is modulated in a period of about six or seven solar cycles. ii) The B-values of the northern and southern hemispheres are also modulated with a similar period to the global one, but iii) they show quasi-oscillatory behavior with a phase shift between them. We examined the yearly fluctuations of the B-values in every solar cycle with reference to the…
We investigate how rotationally-constrained, deep convection might give rise to supergranulation, the largest distinct spatial scale of convection observed in the solar photosphere. While supergranulation is only weakly influenced by rotation, larger spatial scales of convection sample the deep convection zone and are presumably rotationally influenced. We present numerical results from a series of nonlinear, 3-D simulations of rotating convection and examine the velocity power distribution realized under a range of Rossby numbers. When rotation is present, the convective power distribution po…
This paper examines long-term temporal and spatial fluctuations in the solar rotation (more than four solar cycles) by investigating radio emission escapes from various layers of the solar atmosphere during the years 1967-2010. The flux modulation approach can also be used to investigate variations in solar rotation, which is a contentious topic in solar physics. The current study makes use of a time series of radio flux data at different frequencies (245-15400 MHz) obtained at Sagamore Hill Solar Radio Observatory in Massachusetts, USA, and other observatories from 1967 to 2010. The periodici…
The Sun rotates differentially with a fast equator and slow pole. Convection in the solar interior is thought to maintain the differential rotation. However, although many numerical simulations have been conducted to reproduce the solar differential rotation, previous high-resolution calculations with solar parameters fall into the anti-solar (fast pole) differential rotation regime. Consequently, we still do not know the true reason why the Sun has a fast-rotating equator. While the construction of the fast equator requires a strong rotational influence on the convection, the previous calcula…
We carry out a magneto-hydrodynamic (MHD) simulation of convective dynamo in the rotating solar convective envelope driven by the solar radiative diffusive heat flux. The simulation is similar to that reported in Fan & Fang (2014) but with further reduced viscosity and magnetic diffusion. The resulting convective dynamo produces a large scale mean field that exhibits similar irregular cyclic behavior and polarity reversals, and self-consistently maintains a solar-like differential rotation. The main driver for the solar-like differential rotation (with faster rotating equator) is a net outward…
