Influence of temperature profile on solar acoustic modes

Solar global acoustic oscillations in a multilayer model of the solar atmosphere are studied in the plane-parallel geometry. Calculated frequencies of acoustic modes of the Sun are found to depend on parameters of the temperature profile used in the model. Larger influence on frequencies comes from values of the temperature gradient in the convection zone, and less influence from values of the thickness L of the transitional layer and from values of the ratio Tc/Tp of the coronal and the photospheric temperature, Tc and Tp, respectively. The uncertainties in determining these parameters can easily yield frequency shifts that are larger than the observational accuracy. This then indicates a possibility for a diagnostics of solar plasma based on known values of observed oscillation frequencies.     

Local Fractional Frequency Shifts Used as Tracers of Magnetic Activity

Using data from SOI-MDI (Haber et al., 2000), we compute the local frequencies of high-degree p modes and f modes. The frequencies are obtained through ring-diagram mode fitting. The Dense-Pack data set consists of a mosaic of 189 overlapping tiles, each tracked separately at the surface rotation rate over 1664-min time intervals during the Dynamics Programs. Each tile is 16° square and the tile centers are separated by 7.5° in latitude and longitude. For each sampling day and for each tile, we have computed the frequency shift measured relative to the temporal and spatial average of the entire set of frequencies. The motion of active regions as they rotate across the solar disk is vividly traced by these measurements. Active regions appear as locations of large positive frequency shifts. If the shifts are averaged over the solar disk and are scaled down to the appropriate wave number regime, the magnitude and frequency dependence of the shifts are consistent with the measured changes in global oscillation frequencies that occur over the solar cycle. As with the frequency shifts of low-degree global oscillations, the frequency dependence of the shifts indicates that the physical phenomena inducing the shifts is confined to the surface layers of the Sun.     

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

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

Solar oscillations and the equation of state

Summary Accurate measurements of observed frequencies of solar oscillations are providing a wealth of data on the properties of the solar interior. The frequencies depend on solar structure, and on the properties of the plasma in the Sun. Here we consider in particular the dependence on the thermodynamic state. From an analysis of the equations of stellar structure, and the relevant aspects of the properties of the oscillations, we argue that in the convection zone one can isolate information about the equation of state which is relatively unaffected by other uncertainties in the physics of the solar interior. We review the different treatments that have been used to describe the thermodynamics of stellar plasmas. Through application of several of these to the computation of models of the solar envelope we demonstrate that the sensitivity of the observed frequencies is in fact sufficient to distinguish even quite subtle features of the physics of solar matter. This opens up the possibility of using the Sun as a laboratory for statistical mechanics, under conditions that are out of reach in a terrestrial laboratory.     

Properties of r-modes in the Sun

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

Global Effects of Local Sound-Speed Perturbations in the Sun: A Theoretical Study

We study the effect of localized sound-speed perturbations on global mode frequencies by applying techniques of global helioseismology to numerical simulations of the solar acoustic wave field. Extending the method of realization-noise subtraction (e.g., Hanasoge, Duvall, and Couvidat, Astrophys. J. 664, 1234, 2007) to global modes and exploiting the luxury of full spherical coverage, we are able to achieve very highly resolved frequency differences that are then used to study sensitivities and the signatures of the thermal asphericities. We find that i) global modes are almost twice as sensitive to sound-speed perturbations at the bottom of the convection zone in comparison to anomalies well inside the radiative interior (r?0.55R _⊙), ii) the m degeneracy is lifted ever so slightly, as seen in the a coefficients, and iii) modes that propagate in the vicinity of the perturbations show small amplitude shifts. Through comparisons with error estimates obtained from Michelson Doppler Imager (MDI; Scherrer et al., Solar Phys. 162, 129, 1995) observations, we find that the frequency differences are detectable with a sufficiently long time series (70?–?642 days).     

Influence of a chromospheric magnetic field on solar acoustic modes

This paper studies the effect of a magnetic atmosphere on the global solar acoustic oscillations in a simple Cartesian model. First, the influence of the ratio of the coronal and the photospheric temperature τ and the strength of the magnetic field at the base of the corona B_c on the oscillation modes is studied for a convection zone-corona model with a true discontinuity. The ratio τ seems to be an important parameter. Subsequently, the discontinuity is replaced by an intermediate chromospheric layer of thickness L and the effect of the thickness on the frequencies of the acoustic waves is studied. In addition, nonuniformity in the magnetic field, plasma density and temperature in the transition layer gives rise to continuous Alfvén and slow spectra. Modes with characteristic frequencies lying within the range of the continuum may resonantly couple to Alfvén and/or slow waves.     

The potential of solar high-degree modes for structure inversion

It is likely that precise and reliable frequencies of high-degree modes will soon be available from the SOI/MDI experiment. Here we examine the ability of such modes (with l>300) to resolve the solar structure in the near-surface region. In particular, we investigate inversions to determine the adiabatic exponent ₁ as a test of the solar equation of state, as well as the potential of such data to constrain the solar envelope helium abundance.     

Measurements of Frequencies of Solar Oscillations from the Mdi Medium-l Program

Inversions of solar internal structure employ both the frequencies and the associated uncertainties of the solar oscillation modes as input parameters. In this paper we investigate how systematic errors in these input parameters may affect the resulting inferences of the sun's internal structure. Such systematic errors are likely to arise from inaccuracies in the theoretical models which are used to represent the spectral lines in the observational power spectra, from line blending, from asymmetries in the profiles of these lines, and from other factors. In order to study such systematic effects we have employed two different duration observing runs (one of 60 days and the second of 144 days) obtained with the Medium-l Program of the Michelson Doppler Imager experiment onboard the SOHO spacecraft. This observing program provides continuous observations of solar oscillation modes having angular degrees, l, ranging from 0 to ∼ 300. For this study intermediate- and high-degree p-mode oscillations having degrees less than 251 were employed. In the first of our tests we employed two different methods of estimating the modal frequencies and their associated uncertainties from the 144-day observational power spectra. In our second test we also repeated both methods of frequency estimation on the 60-day time series in order to assess the influence of the duration of the observed time series on the computed frequencies and uncertainties. In a third test we investigated the sensitivity of the computed frequencies to the choice of initial-guess, or ‘seed’ frequencies that are used in the frequency estimation codes. In a fourth test we attempted to investigate the possible systematic frequency errors which are introduced when the observational asymmetry in the p-mode peaks is ignored. We carried out this particular test by fitting simple models of asymmetric line profiles to the peaks in the observational power spectra. We were then able to compute the differences between those frequencies and our previous frequencies which had been obtained using the assumption that all of the observational peaks were symmetric in shape. In order to study the possible influence of the two different frequency estimation methods upon the radial profile of the internal sound speed, we carried out four parallel structural inversions using the different sets and subsets of frequency estimates and uncertainties as computed from the 144-day observing run as inputs. The results of these four inversions confirm the previous finding by the GONG project (Gough et al., 1996) and by the MDI Medium-l Program (Kosovichev et al., 1997) that, in a thin layer just beneath the convection zone, helium appears to be less abundant than predicted by theory. However, differences in our four inverted radial sound speed profiles demonstrate that the currently-available techniques for determining the frequencies of the Medium-l oscillation peaks introduce systematic errors which are large enough to affect the results of the structural inversions. Moreover, based upon the differences in these four inverted sound speed profiles, it appears that the choice of which subset of modes is included in a particular inversion and which modes are not included may also be introducing systematic errors into our current understanding of solar internal structure. Hence, it appears to be very important that consistent sets of modal selection criteria be employed. Finally, at least one of the two frequency estimation codes which we used was not sensitive to changes in the input ‘seed’ frequencies which were employed as initial guesses for that code. This result allays fears that the difference in the helium abundance between the sun and the reference solar model in the thin layer beneath the convection zone which was mentioned above might have been due to the particular seed frequencies which were employed in the earlier inversions. Since this thin layer may likely be the place where the solar dynamo operates, it will be extremely important to observe any possible evolution of this transition layer throughout the upcoming 11-year activity cycle. Supplementary material to this paper is available in electronic form at http://dx.doi.org/10.1023/A:1004963425123     

Periodic solar EUV flux monitored near Venus

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

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

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

On the Excitation Mechanism of Solar Five-minute Oscillations

The excitation mechanism of solar five-minute oscillations is studied in the present paper. We calculated the non-adiabatic oscillations of low- and intermediate-degree (l = 1 − 25) g4-p39 modes for the Sun. Both the thermodynamic and dynamic couplings are taken into account by using our non-local and time-dependent theory of convection. The results show that all the lowfrequencyf- and p-modes with periods P > 5.4 min are pulsationally unstable, while the coupling between convection and oscillations is neglected. However, when the convection coupling is taken into account, all the g- and low-frequency f- and p-modes with periods longer than 16 minutes (except the low-degree p1-modes) and the high frequency p-modes with periods shorter than 3 minutes become stable, and the intermediate-frequency p-modes with period from 3 to 16 minutes are pulsationally unstable. The pulsation amplitude growth rates depend only on the frequency and almost do not depend on l. They achieve the maximum at ν 3700 μHz (or P 270 sec). The coupling between convection and oscillations plays a key role for stabilization of low-frequency f- and p-modes and excitation of intermediate-frequency p-modes. We propose that the solar 5-minute oscillations are not caused by any single excitation mechanism, but they are resulted from the combined effect of “regular” coupling between convection and oscillations and turbulent stochastic excitation. For low- and intermediatefrequency p-modes, the coupling between convection and oscillations dominates; while for high-frequency modes, stochastic excitation dominates.     

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).     

Diamagnetic boundary layers: A kinetic theory

We present a kinetic theory for boundary layers associated with MHD tangential discontinuities in a collisionless magnetized plasma such as those observed in the solar wind. The theory consists of finding self-consistent solutions of Vlasov's equation and Maxwell's equation for stationary, one-dimensional boundary layers separating two Maxwellian plasma states. Layers in which the current is carried by electrons are found to have a thickness of the order of a few electron gyroradii, but the drift speed of the current-carrying electrons is found to exceed the Alfvén speed, and accordingly such layers are not stable. Several types of layers, in which the current is carried by protons are discussed; in particular, we considered cases in which the magnetic field intensity and/or direction changed across the layer. In every case, the thickness was of the order of a few proton gyroradii and the field changed smoothly, although the characteristics depended somewhat on the boundary conditions. The drift speed was always less than the Alfvén speed, consistent with stability of such structures. Our results are consistent with the observations of boundary layers in the solar wind near 1 AU.     

Heat flow near obstacles in the solar convection zone

Disturbances in the heat flow in the solar convection zone are calculated with a turbulent thermal diffusion coefficient based on a mixing length approximation. As a consequence of the radiative boundary condition at the surface and the strong increase of the diffusion coefficient with depth, the convection zone resembles a thermally superconducting shell enclosed between a thin surface layer and an interior core of low thermal conductivity. Thermal disturbances originating in the convection zone do not penetrate into the interior, and penetrate only weakly through the solar surface. A thermally isolating obstacle buried entirely in the convection zone casts a shadow of reduced temperature at the solar surface; the brightening surrounding this shadow is undetectable. The shadow is weak unless the object is located close to the surface (less than 2000 km). Assuming a sunspot to be an area of reduced thermal conductivity which extends a finite depth into the convection zone, the heat flow around this obstacle is calculated. The heat flux blocked below the spot (missing flux) spreads over a very extended area surrounding the spot. The brightening corresponding to this missing flux is undetectable if the reduction of the thermal conductivity extends to a depth greater than 1000 km. It is concluded that no effect other than a decrease of the convective efficiency is needed to explain the temperature change observed at the solar surface in and around a sunspot. The energy balance is calculated between magnetic flux tubes, oriented vertically in the solar surface, (magnetic elements in active regions and the quiet network) and their surroundings. Near the visible surface radiation enters the tube laterally from the surrounding convection zone. The heating effect of this influx is important for small tubes (less than a few arcseconds). Due to this influx tubes less than about 1 in diameter can appear as bright structures irrespective of the amount of heat conveyed along the tube itself. Through the lateral influx, small tubes such as are found in the quiet network act as little leaks in the solar surface through which an excess heat flux escapes from the convection zone.     

The effects of non-sphericity in diagnosis of solar and stellar atmospheres

Between the interplanetary medium, filled by winds, magnetic structures, etc., and the interior of stars, opaque, and dominated heavily by the gravitational spherical field, the stellar atmosphere is a place where the true physical equilibrium, on the inside, sufficiently described by the parameters L, M, R, and the chemical composition X, Y, Z, is progressively changing into a situation far for equilibrium, which needs many more parameters to be properly described.The assumption that the equilibrium situation was dominating in the atmosphere has been generally accepted during the first half of this century. Since 1950 or so, we progressively learnt that the thermodynamical equilibrium (TE), and even the local thermodynamical equilibrium (LTE), are far from being actually in existence, that the radiative equilibrium (RE) is not actually perfect, convection, diffusion, magnetism, dissipation processes... playing a non-negligible part in the energy transport, that the hydrostatic equilibrium (HE) is only an approximation, as the convection and the magnetism are affecting the atmospheric layers, that neither the sphericity of atmospheric layers (plane-parallel hypothesis: PP) is achieved, nor the homogeneity of stellar iso- layers. During the 1950s and following decades, we began to suspect these difficulties and their consequences. In this paper, we turn towards a new consequence of the last-mentioned effect: the influence of non-sphericity and inhomogeneity upon the stellar (and solar perhaps) abundances of elements.Dedicated to Cornelis de Jager     

Thermal convection instability of a two-component fluid layer in hydromagnetics

The thermal convection instability of a two component fluid layer subjected to a temperature gradient has been studied in the presence of an applied magnetic field. The associated thermal diffusion separation has a predominant effect even when the separations are small. Solutions for the non-oscillatory marginal states have been obtained. It is shown that the concentration gradient has a stabiliting or destabilizing effect according as T<or>0. Approximate solutions for the oscillatory solutions have been obtained by the method of variational principle and the dispersion relation has been solved numerically.     

Intermediate degree p-mode frequency splittings near solar maximum

We report on observations of global solar Ca K-line intensity oscillations taken in May 1991 from Mees Solar Observatory, Hawaii. We measurep-mode frequency splittings for modes of spherical harmonic degrees between 20 and 129 averaged over the radial order of the modes. Our measurement of the antisymmetric component of the splittings is comparable with previous measurements and thus indicates a decrease in the latitudinal differential rotation with depth into the convection zone and the upper radiative zone. We find evidence for a 1% variation in the rotation rate of the upper convection zone roughly in phase with the solar activity cycle. Our measurement of the symmetric component of the splittings is of the same order as was reported from the previous solar maximum and is an order of magnitude larger than has been measured near solar minimum. From the degree dependence of the symmetric component of the splittings, we find evidence for an aspherical fractional sound speed perturbation located at a depth of 0.85 ± 0.05 solar radii. This perturbation has a magnitude ofc/c +9 × 10^–4 at the equator relative to the poles. Additionally, there is evidence for a near-surface aspherical sound speed perturbation of smaller magnitudec/c +4 × 10^–4 at the equator relative to the poles. If an intense global magnetic field were the dominant source of the observed symmetric component of the splittings, instead of latitudinal gradients in the sound speed, then global fields of order 10⁵ G would be required in the convection zone.     

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.     

Linear hydrodynamical equations coupled with radiative transfer in a non-isothermal atmosphere

A method coupling the hydrodynamical equations and radiative transfer in a realistic solar model atmosphere is described. The influence of the temperature gradient of the model and the radiative dissipation is pointed out.The effect of the large temperature gradient is important in the layers where the optical depth ₅₀₀₀ is greater than 0.5; the ratio between the amplitude of the temperature and the velocity fluctuations decreases with the altitude by a factor 2 between = 1 and = 0.5 and in the case of the acoustic waves, the phase shift between these fluctuations is small.The radiative energy loss in the thick layers ( ₅₀₀₀ = 1) leads to a decrease of the vertical phase velocity of the waves and to a damping of their amplitudes in the layers of intermediate optical depth (10^-2 < ₅₀₀₀ < 0.5). The effect of the dissipation is negligible in the thin layers (₅₀₀₀ < 10^-2).