Seismology of stellar envelopes: probing the outer layers of a star through the scattering of acoustic waves

The outer layers of Sun-like stars are regions of rapid spatial variation which modulate the p-mode frequencies by partially reflecting the constituent acoustic waves. With the accuracy that has been achieved by current solar observations, and that is expected from imminent stellar observations, this modulation can be observed from the spectra of the low-degree modes. We present a new and simple theoretical calculation to determine the leading terms in an asymptotic expansion of the outer phase of these modes, which is determined by the structure of the surface layers of the star. Our procedure is to compare the stellar envelope with a plane-parallel polytropic envelope, which we regard as a smooth reference background state. Then we can isolate a seismic signature of the acoustic phase and relate it to the stratification of the outer layers of the convection zone. One can thereby constrain theories of convection that are used to construct the convection zones of the Sun and Sun-like stars. The accuracy of the diagnostic is tested in the solar case by comparing the predicted outer phase with an exact numerical calculation.     

Solar p and g modes in a model with a convection layer above a sub-adiabatic interior

We study high-order acoustic modes which reside in the outer layers of the solar interior. Magnetic field effects are not taken into account in this paper as we wish first to filter out how the modal frequencies depend on physical characteristics of a particular model structure of the Sun. In particular, we are interested in how the modal frequencies of solar global oscillations depend on the thickness of the convection layer and on the temperature gradient of the solar interior below. The model we use consists of three planar layers: an isothermal atmosphere, while the convection layer and the interior have temperature gradients that are adiabatic and sub-adiabatic, respectively. The presence of a convection layer with a finite thickness brings in additional modes while the variations in temperature gradient of the interior cause shifts in eigenfrequencies that are more pronounced for the p modes than for the g modes. These shifts can easily be of the order of several hundreds of Hz, which is much larger than the observational accuracy.     

Solar Activity and Deep Convection Modeling

We suggest from synoptic charts of radial magnetic field and intensities of spectral lines (Fe?i, He?ii, and Fe?ix/x) over Carrington rotations 1942??C?2050 that deep convective layers control the pattern of large-scale solar activity. A new result is a Kolmogorov-type energy spectrum of the longitudinal variations of solar activity. This spectrum for nonphotospheric scales of convection (harmonic number m<100) is a new ??fingerprint?? of turbulence in the deep layers of the solar convection zone (CZ). The manifestation of one source of convective turbulence in the deep CZ is revealed as the excess in the power spectrum over the Kolmogorov spectrum. This source may be identified with giant convection cells at the CZ bottom. The convective cascade of the turbulence starts at the vortex size corresponding to the trans-CZ convective cells with the turnover time which the mixing length theory (MLT) predicts. This connection between the MLT formalism and real features in the Sun could account for the success of the MLT in stellar modeling.     

Deep Convection and the Solar Chromosphere

The manifestation of convection in deep layers of the Sun has been found in the dynamics of solar surface activity (Arkhypov, Antonov, and Khodachenko in Solar Phys. 270, 1, 2011). Some chromospheric phenomena could be connected with deep convection, too. We justify this hypothesis with sunspot, Ca ii, Hα, and millimeter-wave radio data. It is argued that large-scale (20 to 25 deg) bright regions in the chromosphere, surrounded by dark halos with diameters of 40° to 50°, can be manifestations of giant convection cells. The ascending and descending flows in such cells modulate the emergence of magnetic tubes generating the high-temperature regions and low-temperature halo in the chromosphere. Our estimates of the rotation rate of such features confirm their association with deep (≳ 35 Mm) layers of the solar convection zone.     

A model of the solar convection zone

A model of the convection zone is presented which matches an empirical model atmosphere (HSRA) and an interior model. A mixing length formalism containing four adjustable parameters is used. Thermodynamical considerations provide limits on two of these parameters. The average temperature-pressure relation depends on two or three combinations of the four parameters. Observational information on the structure of the outermost layers of the convection zone, and the value of the solar radius limit the range of possible parameter combinations. It is shown that in spite of the remaining freedom of choice of the parameters, the mean temperature-pressure relation is fixed well by these data.The reality of a small density inversion in the HSRA model is investigated. The discrepancy between the present model and a solar model by Mullan (1971) is discussed briefly.     

Helioseismology and the solar interior dynamics

The inversion of helioseismic modes leads to the sound velocity inside the Sun with a precision of about 0.1 per cent. Comparisoons of solar models with the &amp;#x201C;seismic sun&amp;#x201D; represent powerful tools to test the physics: depth of the convection zone, equation of state, opacities, element diffusion processes and mixing inside the radiative zone. We now have evidence that microscopic diffusion (element segregation) does occur below the convection zone, leading to a mild helium depletion in the solar outer layers. Meanwhile this process must be slowed down by some macroscopic effect, presumably rotation-induced mixing. The same mixing is also responsible for the observed lithium depletion. On the other hand, the observations of beryllium and helium 3 impose specific constraints on the depth of this mildly mixed zone. Helioseismology also gives information on the internal solar rotation: while differential rotation exists in the convection zone, solid rotation prevails in the radiative zone, and the transition layer (the so-called &amp;#x201C;tachocline&amp;#x201D;) is very small. These effects are discussed, together with the astrophysical constraints on the solar neutrino fluxes.     

Horizontal convective velocity field obtained from the observations of the solar limb

Structure of horizontal convective currents in the solar atmosphere has been investigated using profiles of the λ ≈ 532.42 nm neutral iron line which were observed at the solar limb with high spatial resolution. The asymmetry of the observed line was shown to arise when approaching the solar limb. The spatial and time velocity variations were simulated using the λ-meter technique. Acoustic waves were removed using the k-ω filters. The convection currents on various spatial scales were distinguished, namely, those connected with granulation, mesogranulation, and supergranulation. The spatial and time distribution of the convection velocities in the photosphere and in the low chromosphere has been analyzed. The horizontal currents were shown to exist on granulation, mesogranulation, and supergranulation scales as low as h ≈ 250 km, and the granulation and mesogranulation horizontal velocities increase with height. In the photospheric layers, the supergranulation vertical-velocity field appears almost invariable, while the supergranulation horizontal-velocity field can vary with height. The horizontal velocity distribution within large convection currents is found to be asymmetric on granulation, mesogranulation, and supergranulation scales.     

On the stability of mean-field models of the solar convection zone

The stability of the solutions of the mean-field theories of turbulent media is questioned. It is done here for the model equations for the solar convection zone which have been used, in particular, to explain the differential rotation. We present an approximation valid for axisymmetric, short-wave disturbances. A critical local Rayleigh number can be defined - involving eddy diffusivities - above which the stratification becomes unstable. For mixing-length models of the solar convection zone we always find sub-critical Rayleigh numbers. One must be careful, however, with other theoretical models. Those we have considered do not reach sufficiently high surface pressure values so that there the associated Rayleigh numbers exceed their critical limits. In the outermost layers in such models, therefore, the solutions could really be unstable.     

SOLAR CONSTANT TEMPORAL AND FREQUENCY CHARACTERISTICS

The Sun's total irradiance at the mean Sun-Earth distance decreased from mid-1979 to mid-1987 during the descending part of solar cycle 21. After the minimum had been reached it increased with the onset of cycle 22 and came to a maximum at mid-1991 during the highest solar activity of cycle 22. From the modelized shape of the time signal of the solar constant based on the Space Absolute Radiometric Reference (SARR), temporal, amplitude and behaviour characteristics are derived. It is suggested that the variation observed over a period of more than 14 years is the response of the outer solar layers, the photosphere in particular, to some excitation originating somewhere near the bottom of the solar convection zone also responsible for the solar spots and the correlated photospheric features. Wavelet analysis and periodiograms are shown for the solar constant and the sunspot index. Their non-stationarity is well illustrated as well as strong recurrent periods.     

Studying magneto‐convection by numerical simulation

Following a brief overview of the two main approaches to investigate the interaction between magnetic fields and convective flows near the solar surface layers by numerical simulation, namely idealized model problems and ‘realistic’ large‐eddy simulations, we present first results obtained with a newly developed MHD code. The first example concerns the realistic simulation of the magnetic field dynamics in a solar plage region while the second example demonstrates small‐scale dynamo action in idealized compressible convection.     

The coupling of solar convection and rotation – (Invited Review)

In recent years, helioseismology has provided an unprecedented look at the dynamics of the solar interior. These new insights have been accompanied by tremendous advances in high-performance computing technology, prompting increasingly sophisticated and realistic numerical models of solar convection. Among the most important helioseismic constraints on global-scale convection models is the mean differential rotation profile of the solar envelope, which is established by convection under the influence of rotation. The highly turbulent nature of solar convection makes this rotational influence difficult to determine and model. I will begin this review by discussing the solar rotation profile inferred from helioseismic measurements and various theoretical and numerical approaches to account for it. Computational constraints limited early numerical models to relatively laminar flow regimes but more recent investigations have begun to explore the distinct nature of turbulent convection. After a brief overview of empirical and numerical results on the related Rayleigh-Bernard system, I will outline the current state of numerical modeling of turbulent convection in rotating, stratified fluids, first in Cartesian and then in spherical geometries. The emphasis throughout will be on how rotation influences the structure, evolution, and transport processes of turbulent convection and what type of differential rotation can result.     

Testing turbulent convection theory in solar models – I. Structure of the solar convection zone

Turbulent convection models (TCMs) based on hydrodynamic moment equations are compared with the classical mixing-length theory (MLT) in solar models. The aim is to test the effects of some physical processes on the structure of the solar convection zone, such as the dissipation, diffusion and anisotropy of turbulence that have been ignored in the MLT. Free parameters introduced by the TCMs are also tested in order to find appropriate values for astrophysical applications. It is found that the TCMs usually give larger convective heat fluxes than the MLT does, and the heat transport efficiency is sensitively related to the dissipation parameters used in the TCMs. As a result of calibrating to the present solar values, our solar models usually have rather smaller values of the mixing length to local pressure scaleheight ratio than the standard solar model. The turbulent diffusion is found to have important effects on the structure of the solar convection zone. It leads to significantly lowered and expanded profiles for the Reynolds correlations, and a larger temperature gradient in the central part of the superadiabatic convection region but a smaller one near the boundaries of the convection zone. It is interesting to note that, due to a careful treatment of turbulence developing towards isotropic state, our non-local TCM results in radially dominated motion in the central part and horizontally dominated motion near the boundaries of the convection zone, just as what has been observed in many 3D numerical simulations. Our solar models with the TCMs give small but meaningful differences in the temperature and sound speed profiles compared with the standard solar model using the MLT.     

An Exploration of Non-kinematic Effects in Flux Transport Dynamos

Recent global magnetohydrodynamical simulations of solar convection producing a large-scale magnetic field undergoing regular, solar-like polarity reversals also present related cyclic modulations of large-scale flows developing in the convecting layers. Examination of these simulations reveal that the meridional flow, a crucial element in flux transport dynamos, is driven at least in part by the Lorentz force associated with the cycling large-scale magnetic field. This suggests that the backreaction of the field onto the flow may have a pronounced influence on the long-term evolution of the dynamo. We explore some of the associated dynamics using a low-order dynamo model that includes this Lorentz force feedback. We identify several characteristic solutions which include single period cycles, period doubling and chaos. To emulate the role of turbulence in the backreaction process we subject the model to stochastic fluctuations in the parameter that controls the Lorentz force amplitude. We find that short term fluctuations produce long-term modulations of the solar cycle and, in some cases, grand minima episodes where the amplitude of the magnetic field decays to near zero. The chain of events that triggers these quiescent phases is identified. A subsequent analysis of the energy transfer between large-scale fields and flows in the global magnetohydrodynamical simulation of solar convection shows that the magnetic field extracts energy from the solar differential rotation and deposits part of that energy into the meridional flow. The potential consequences of this marked departure from the kinematic regime are discussed in the context of current solar cycle modeling efforts based on flux transport dynamos.     

Anisotropy and Dissipation of Turbulence and Their Effects on Solar Models

1 INTRODUCTIONThe maing-length theory (MLT) is the most commonly used approach to calculate convective energy transport in stars and other astrophysical situations. Based on the original idea ofPrandtl (1952) that turbulent parcels trallsfer heat in a similar way as molecules of gas do inthermal conduction, the MLT assumes that convection cells, drived by buoyancy, move thlougha ~ng length 1 and release the heat they carry when they merge with their environment. Themost widely adopted f…     

Radiative Cooling in MHD Models of the Quiet Sun Convection Zone and Corona

We present a series of numerical simulations of the quiet-Sun plasma threaded by magnetic fields that extend from the upper convection zone into the low corona. We discuss an efficient, simplified approximation to the physics of optically thick radiative transport through the surface layers, and investigate the effects of convective turbulence on the magnetic structure of the Sun’s atmosphere in an initially unipolar (open field) region. We find that the net Poynting flux below the surface is on average directed toward the interior, while in the photosphere and chromosphere the net flow of electromagnetic energy is outward into the solar corona. Overturning convective motions between these layers driven by rapid radiative cooling appears to be the source of energy for the oppositely directed fluxes of electromagnetic energy.     

Sensitivity of the sub-photospheric flow fields inferred from ring-diagram analysis to the change on the solar model

We study the effect of the change of solar model parameters on the measurements of the horizontal velocity flow components based on the analysis of high-degree modes using the ring-diagram local helioseismic technique. We show that changing the equation of state, opacities, surface heavy-element abundances or the modeling of convection do not affect the sub-photospheric flow field measurements. However, the modeling of outermost layers can affect the measurements if an important amount of high radial order modes (high-frequency modes) are included in the analysis.     

Adiabatic vibrational frequencies of local and nonlocal convection models of the sun

The eigen-vibrational frequencies of Xiong Da-run's nonlocal and local convection models of solar envelope are calculated and compared. The differences between the observational and theoretical vibrational frequencies are less than 1%. They can be divided into two isolated groups. For modes with l ≥ 60, all the differences between observed and theoretical eigen-vibrational frequencies are distributed in a narrow and inclined belt in the (Δv ≈ v)-diagram. This shows that the theoretical model of solar convective region can approximately reflect the intrinsic structure of the sun in the region of r = (0.70–0.95)R. The discrepancies between the theoretical and observational frequencies come from the outer layers. For modes with l < 60, the theoretical vibrational frequency is smaller than the observational one. This implies that the temperature of the upper part of the convectively unstable region is rather low. The frequency difference is more dispersed in the local convection model than in the nonlocal convection model. For the intermediate- and low-frequency ranges (v < 3000), the difference between the two models is small, while for the high-frequency range (v ≥ 3000) the frequency in the local model is higher than in the nonlocal model. This means that the temperature of the radiation region beneath the convective region is higher in the local convection model than in the nonlocal convection model. The nonlocal model is nearer to the observation than the local model.     

Influence of surface layers on the seismic estimate of the solar radius

Possible systematic errors in determining the solar radius from the f-mode frequencies are studied to find that the input physics governing the structure of outermost layers of the Sun has significant influence on the estimated radius. It is investigated how treatment of convection and low temperature opacity may influence the determination of radius and it is found that it may be changed by as much as 100 km. The best estimate of the solar radius is found to be 695.77±0.1 Mm, where the error bars represent estimate of systematic errors, while the statistical errors are very small (1 km).     

The structure of the solar convective overshooting zone

Using a non-local theory of convection, we calculated the structure of the solar convection zone, paying special attention to the detailed structure of the lower overshooting zone. Our results show that an extended transition zone exists near the bottom of the convection zone, where the temperature gradient turns smoothly from adiabatic in the convection zone to radiative in solar interior. A super-radiative temperature region is found in the overshooting zone under the solar convection zone, where     ,     ,     and     . The extension of the super-radiative region (defined by     l is about 0.63  H _P (0.053 R_⊙). A careful comparison of the distribution of adiabatic sound speed and density with the local one is carried out. It is found, strikingly, that the distribution of adiabatic sound speed and density of our model is roughly consistent with the results of reversion from solar oscillation observations.