Self-excitation of motions in near-convection zones and the generation of solar magnetic field

In view of the recently discovered time variations in rotation velocity within the solar differentially rotating tachocline (Howe et al. 2000), we study conditions for the equilibrium and excitation of motions in nonrigidly rotating magnetized layers of the radiative zones located near the boundaries of the convection zone. The emphasis is on the possible relationship between quasi-periodic tachocline pulsations and the generation of a nonaxisymmetric magnetic field in the convection zone. This field generation is studied under the assumption that it results from a reduction in the expenditure of energy on convective heat transport. The (antisymmetric about the equator) field is shown to increase in strength if there are both a radial gradient in angular velocity and steady-state axisymmetric meridional circulation of matter. The sense of circulation is assumed to change (causing the sign of the generated field to change) after the maximum permissible field strength is reached. This is apparently attributable to the excitation of the corresponding turbulent viscosity of the medium. It is also important that the cyclic field variations under discussion are accompanied by variations in solar-type dipole magnetic field.     

On the effect of magnetic field on the low-l solar p-mode oscillations

One of the possible magnetic field effects on the stellar pulsations is known to be a splitting in the observed frequencies. Using this knowledge in the solar convection zone, there are two aims in this work Considering the Sun as an incompressible fluid, our first objective was to investigate the variation of the physical parameters in the 30% outermost convective solar layer, during a pulsation period. The second purpose was to calculate, by means of the spherical harmonics, the shifts on the low-l p-mode frequencies which could be caused by the presence of the magnetic field in the Sun. The first order perturbation approximation was used in order to calculate analytically the resulting frequency shifts and the small perturbations on the magnetic field, as well as the physical parameters, such as density, pressure and temperature, of a Standard Solar Model excluding both rotation and magnetic field (Christensen-Dalsgaard et al., 1996) in the unperturbed equilibrium case.     

A new inversion of solar rotational splitting data

The fine structure of the solar p-mode spectrum is used to obtain an estimate of the Sun's internal rotation rate, as a function of both latitude and depth, for fractional radii in the range 0.55 < r/R < 0.85. Because each piece of data is a weighted average of the rotation rate over an extended region of the solar interior and because the number of such measurements is finite, such an inversion unavoidably has limited resolution and suffers from other systematic errors. Accordingly, the results of the inversion presented here and other published inversions should be interpreted with this in mind. These problems are discussed in detail for the present inversion.Such systematic errors can be avoided by seeking not a functional form of the rotation rate but rather the values of suitably weighted averages of the rotation rate (or of functions derivable from it). Here it is shown that the value of a broad average over depth of the radial gradient of the solar rotation rate, concentrated in the convection zone, is consistent with the gradient being zero in this region but is apparently inconsistent with the picture of constant rotation on cylindrical surfaces aligned with the rotation axis. This result, which confirms the inference from the former type of inversion, is of importance for modelling the dynamics of the convection zone and also for dynamo models of the Sun's magnetic field.The National Center for Atmospheric Research is supported by the National Science Foundation.     

Compressible Fluid Model for the Seismic Waves Generated by A Sunquake

The solar convection zone is modeled as a horizontally stratified atmosphere with a constant gravitational field and an adiabatic temperature gradient (a neutrally stratified polytrope). At equilibrium, the gas pressure and density decreases to zero at the solar surface so that the solar surface is treated as a free surface which is bounded by vacuum. The evolution of small amplitude perturbations about the equilibrium state is described by the linearized Euler equations for an inviscid compressible fluid with an adiabatic equation of state. A sunquake is initiated at time zero by means of an initial perturbation with a Gaussian velocity profile and the exact solution of the initial value problem is obtained in terms of a Fourier integral. Comparisons between theory and observations indicate that this highly simplified model is able to predict the propagation of sunquake waves across the solar surface with an error of approximately 10% or 20%.     

Differential rotation and meridional flow of Arcturus

The spectroscopic variability of Arcturus hints at cyclic activity cycle and differential rotation. This could provide a test of current theoretical models of solar and stellar dynamos. To examine the applicability of current models of the flux transport dynamo to Arcturus, we compute a mean‐field model for its internal rotation, meridional flow, and convective heat transport in the convective envelope. We then compare the conditions for dynamo action with those on the Sun. We find solar‐type surface rotation with about 1/10th of the shear found on the solar surface. The rotation rate increases monotonically with depth at all latitudes throughout the whole convection zone. In the lower part of the convection zone the horizontal shear vanishes and there is a strong radial gradient. The surface meridional flow has maximum speed of 170 m/s and is directed towards the equator at high and towards the poles at low latitudes. Turbulent magnetic diffusivity is of the order 10¹⁵–10¹⁶ cm²/s. The conditions on Arcturus are not favorable for a circulation‐dominated dynamo (© 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A joined model for solar dynamo and differential rotation

A model for the solar dynamo, consistent in global flow and numerical method employed with the differential rotation model, is developed. The magnetic turbulent diffusivity is expressed in terms of the entropy gradient, which is controlled by the model equations. The magnetic Prandtl number and latitudinal profile of the alpha-effect are specified by fitting the computed period of the activity cycle and the equatorial symmetry of magnetic fields to observations. Then, the instants of polar field reversals and time-latitude diagrams of the fields also come into agreement with observations. The poloidal field has a maximum amplitude of about 10 Gs in the polar regions. The toroidal field of several thousand Gauss concentrates near the base of the convection zone and is transported towards the equator by the meridional flow. The model predicts a value of about 10³⁷ erg for the total magnetic energy of large-scale fields in the solar convection zone.     

The magnetic field transfer in the solar convective zone

The large-scale azimuth magnetic field is pumping to the bottom of the solar convective zone due to the diamagnetic action of turbulent conductive fluids. When the field at the bottom is of about 10³ G, an equilibrium is established between diamagnetic pumping and buoyancy.If, in addition to the density gradient, an additional anisotropy exists (for instance, due to rotation), another mechanism of the magnetic field transfer appears, the efficiency of which greatly depends on the magnitude of the anistropy parameter.     

Hydrodynamic Source of the 11-year Solar Variations

In the traditional axisymmetric models of the 11-year solar cycle, oscillations of the magnetic fields appear in the background of nonoscillating (over time scale considered) turbulent velocity fields and differential rotation. In this paper, an alternative approach is developed: The excitation of magnetic oscillations with the 22-year period is the consequence of hydrodynamic oscillations with the 11-year period. In the excitation of hydrodynamic oscillations, two processes taking place in high latitudes near the interface between the convective and radiative zones play a key role. One is forcing of the westerly zonal flow, the conditions for which are due to deformation of the interfacial surface. The other process is the excitation of a shear instability of zonal flow as a consequence of a strong radial gradient of angular velocity. The development of a shear instability at some stage brings about the disruption of the forcing of differential rotation. In the first (hydrodynamic) part of the paper, the dynamics of axisymmetric flows near the bottom of the convection zone is numerically simulated. Forcing of differential rotation having velocity shear in latitude and the existence of solutions in the form of torsional waves with the 11-year oscillation period are shown. In the second part the dynamics of the magnetic field is studied. The most pronounced peculiarities of the solutions are the existence of forced oscillations with the 22-year period and the drift of the toroidal magnetic field component from the mid latitudes to the equator. In high and low latitudes after cycle maximum, the toroidal component is of opposite sign in accordance with observations. In the third part, the transport of momentum from the bottom of the convection zone to the outer surface by virtue of diffusivity is considered. The existence of some sources of differential rotation in the convection zone is not implied. A qualitative correspondence of the differential rotation profile in the bulk of the convection zone and on its outer surface to experimental data is shown. The time correspondence between torsional and magnetic oscillations is also in accordance with observations.     

Simulating the generation of the solar toroidal magnetic field by differential rotation

Within the kinematic dynamo theory, we construct a mathematical model for the evolution of the solar toroidal magnetic field, excited by the differential rotation of the convective zone in the presence of a poloidal field of a relic origin. We use a velocity profile obtained by decoding the data of helioseismological experiments. For the model of ideal magnetic hydrodynamics, we calculate the latitudinal profiles of the increasing-with-time toroidal field at different depths in the solar convection zone. It is found that, in the region of differential rotation, the excited toroidal field shows substantial fluctuations in magnitude with depth. Based on the simulations results, we propose an explanation for the “incorrect polarity” of magnetic bipolar sunspot groups in solar cycles.     

A comparison of differential rotation measurements – (Invited Review)

Observers have long measured solar rotation with different techniques and obtained different results. This paper compares differential rotation measurements from four techniques: Doppler shift, Doppler feature tracking, magnetic feature tracking, and p-mode splittings. The different rotation rates measured by the first three techniques are interpreted as rotation rates of solar phenomena which depend on the properties and depth of that which is measured. This interpretation is supported by comparison with rotation measurements obtained from p-mode splittings except for Doppler features. The rotation rate of the plasma corresponds to the surface rate obtained by inversions; the rates of magnetic features correspond to the rotation rate at various depths within the convection zone. Supergranulation rotates at a rate greater than the maximum rotation rate within the convection zone, suggesting that supergranules are not simple convection cells anchored at a particular depth.     

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.     

Coronal general magnetic field evolution as a new parameter of the solar cycle

The magnetic field lines of the corona associated with the solar-cycle surface general magnetic field are calculated by a potential-field approximation to study the solar-cycle evolution of the geometry of the coronal field. The surface field evolution used here is the radial field evolution, predicted by a model of the solar cycle driven by the dynamo action of the global convection, and justified observationally using Mount Wilson magnetic synoptic chart data. The evolution of the calculated coronal general field is now good for comparison with observations and shows the following. (i) The field of the polar and high-altitude corona has dipolar structure in almost all phases of the solar cycle except in a short time interval around maximum phase despite the quadrupolar structure of the general magnetic field at the surface; quadrupolar field forms loop-like structure in the lower corona. The almost-dipolar structure of the polar and high-latitude corona and the loop formation of the equatorial lower corona explain the appearance of the undisturbed minimum corona observed at eclipses. (ii) The polar field lines are directed almost radially at the minimum phase, which should be responsible for polar plumes. The field lines slowly open up to participate in the loop-like structure of the equatorial lower corona, and rapidly change their structure and polarity at the maximum phase, to resume the almost radial configuration slowly, (iii) During the rapidly changing maximum phase, the field lines do not penetrate deep into the interplanetary space resulting in the absence of polar plumes and the appearance of the circular corona- the maximum corona. In this phase, the coronal field should not be approximated by a dipole field. The surface field evolution which can explain such behaviors of the corona is characteristic of the solar-cycle process dominated by the latitudinal gradient of the differential rotation. If the radial gradient dominated in the subsurface process, the coronal evolution would look quite different and would show latitudinal propagation of enhancement of activity. Although nonaxisymmetric features should be superposed on the axisymmetric general field to express the real corona, the general field can be a basic coronal field in studying long-term interaction between the convection zone and the interstellar space especially in studying the magnetic braking of the solar rotation.     

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.     

Evidence for solar neutrino flux variability and its implications

Although KamLAND apparently rules out resonant-spin-flavor-precession (RSFP) as an explanation of the solar neutrino deficit, the solar neutrino fluxes in the Cl and Ga experiments appear to vary with solar rotation. Added to this evidence, summarized here, a power spectrum analysis of the Super-Kamiokande data reveals significant variation in the flux matching a dominant rotation rate observed in the solar magnetic field in the same time period. Three frequency peaks, all related to this rotation rate, can be explained quantitatively. A Super-Kamiokande paper reported no time variation of the flux, but showed the same peaks, there interpreted as statistically insignificant, due to an inappropriate analysis. This modulation is small (7%) in the Super-Kamiokande energy region (and below the sensitivity of the Super-Kamiokande analysis) and is consistent with RSFP as a subdominant neutrino process in the convection zone. The data display effects that correspond to solar-cycle changes in the magnetic field, typical of the convection zone. This subdominant process requires new physics: a large neutrino transition magnetic moment and a light sterile neutrino, since an effect of this amplitude occurring in the convection zone cannot be achieved with the three known neutrinos. It does, however, resolve current problems in providing fits to all experimental estimates of the mean neutrino flux, and is compatible with the extensive evidence for solar neutrino flux variability.     

Maunder convection mode on the Sun and long solar activity minima

A model of velocity field oscillations in the solar convective zone is suggested. The system of convective equations is investigated for a thin rotating spherical envelope when the rotation velocity is depended on the coordinates. It is shown that two different structures of convective cells (longitudinal, or latitudinal) can exist in the envelope depending on gradients values of the rotation velocity and Prandtl number. It is supposed that two different regimes of convection (stationary and autofluctuating) are possible in the envelope when the angular velocity gradients are determined by the convection itself. In the case of autofluctuating regime the alternation of longitudinal and latitudinal structure of convection is realized. If one assumes that on the Sun there exists an autooscillating convection regime, then the periods of the existence of latitudinal convection structure may be associated with long periods of activity minima since according to Cowling's theorem, the action of the axisymmetric magnetic field generation mechanism is impossible under conditions of axisymmetric velocity structures.     

Evidence for solar neutrino flux variability and its implications

Although KamLAND apparently rules out resonant-spin-flavor-precession (RSFP) as an explanation of the solar neutrino deficit, the solar neutrino fluxes in the Cl and Ga experiments appear to vary with solar rotation. Added to this evidence, summarized here, a power spectrum analysis of the Super-Kamiokande data reveals significant variation in the flux matching a dominant rotation rate observed in the solar magnetic field in the same time period. Three frequency peaks, all related to this rotation rate, can be explained quantitatively. A Super-Kamiokande paper reported no time variation of the flux, but showed the same peaks, there interpreted as statistically insignificant, due to an inappropriate analysis. This modulation is small (7%) in the Super-Kamiokande energy region (and below the sensitivity of the Super-Kamiokande analysis) and is consistent with RSFP as a subdominant neutrino process in the convection zone. The data display effects that correspond to solar-cycle changes in the magnetic field, typical of the convection zone. This subdominant process requires new physics: a large neutrino transition magnetic moment and a light sterile neutrino, since an effect of this amplitude occurring in the convection zone cannot be achieved with the three known neutrinos. It does, however, resolve current problems in providing fits to all experimental estimates of the mean neutrino flux, and is compatible with the extensive evidence for solar neutrino flux variability.     

On convection in the sun

Convective motions driven by a superadiabatic temperature gradient in a viscous thermally conductive medium are considered. Approximate linearized equations governing the perturbation are derived under the following conditions: (i) The ratio of the excess temperature gradient over the adiabatic gradient is small compared with the gradient itself, (ii) The perturbation is of low-frequency type, (iii) The rotation is slow. Only the convective mode is described by these equations (as in the Boussinesq approximation), and the equations are valid for compressible configurations with any ratio between the scale heights of the equilibrium and perturbed quantities. Results of a numerical calculation of unstable perturbations for configurations with a large density stratification are given. They show that under conditions appropriate for the solar convection zone an extremely strong instability is expected to occur if the mixing length is assumed to be equal to 1.5 times the pressure scale height. The horizontal scale of the instability is intermediate between those of granulation and supergranulation. The larger the mixing length, the smaller the growth rate of the instability, and the larger its horizontal scale. Therefore it seems possible to adjust the mixing length to obtain the characteristics corresponding to those of the solar supergranulation. The possible origin of the granulation as an instability in a subsurface zone, where a local increase in the density scale height takes place, is also discussed. To achieve agreement with observations, it seems necessary to assume that the ratio of the mixing length to pressure scale height is an increasing function of the pressure.     

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.     

The slender solar tachocline: a magnetic model

A model for the sharp transition from differential rotation in the solar convection zone to rigid rotation in the radiative interior is presented. Differential rotation in the radiative zone is shown to be quenched efficiently by an internal magnetic field. The poloidal field amplitude, B₀, is the input parameter for our model which determines the transition layer thickness and the toroidal field strength. It is illustrated analytically and confirmed numerically that a rather small field, B₀ = 10⁻⁴ Gauss, suffices to satisfy the helioseismological restrictions on the depth of the differential rotation penetration below the convection zone. The transition layer thickness decreases further with increasing B₀. The toroidal field amplitude B ≃ 200 Gauss is almost independent of B₀.     

On the solar rotation and activity

The interaction between differential rotation and magnetic fields in the solar convection zone was recently modelled by Brun (2004). One consequence of that model is that the Maxwell stresses can oppose the Reynolds stresses, and thus contribute to the transport of the angular momentum towards the solar poles, leading to a reduced differential rotation. So, when magnetic fields are weaker, a more pronounced differential rotation can be expected, yielding a higher rotation velocity at low latitudes taken on the average. This hypothesis is consistent with the behaviour of the solar rotation during the Maunder minimum. In this work we search for similar signatures of the relationship between the solar activity and rotation determined tracing sunspot groups and coronal bright points. We use the extended Greenwich data set (1878–1981) and a series of full-disc solar images taken at 28.4 nm with the EIT instrument on the SOHO spacecraft (1998–2000). We investigate the dependence of the solar rotation on the solar activity (described by the relative sunspot number) and the interplanetary magnetic field (calculated from the interdiurnal variability index). Possible rotational signatures of two weak solar activity cycles at the beginning of the 20th century (Gleissberg minimum) are discussed. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)