Mode Mixing by a Shallow Sunspot

Sunspots are strong absorbers of f and p modes. A possible absorption mechanism is direct conversion to slow magnetoacoustic waves. Calculations based on vertical magnetic field models show that this works well for f modes, but is inadequate for p modes. Using a very simple shallow spot model, in which the effects of the magnetic field are accounted for solely by a surface condition, we investigate the possibility that p modes first scatter into f modes inside the spot, which are then more susceptible to conversion to slow modes. We find that the coupling between an incident p mode and the internal f mode is unlikely to be strong enough to account for the observed absorption, but that the incident modes do couple strongly to the acoustic jacket in some cases, leading to a region immediately around the sunspot where a significant fraction of the surface velocity is due to the jacket modes.     

Magnetothermal instabilities in coronal arcades

The normal mode spectrum for the linearized MHD equations is investigated for a plasma in a cylindrical equilibrium. The equations describing these normal modes are solved numerically using a finite element code. The ballooning equations that describe localized modes are manipulated and a dispersion relation derived. It is shown that as the axial wave numberk is increased, the fundamental thermal and Alfvén modes can coalesce to form overstable magnetothermal modes. The ratio between the magnetic and thermal terms is varied and the existence of the magnetothermal modes examined. The corresponding growth rates are predicted by a WKB solution to the ballooning equations. The existence of these magnetothermal modes may be significant in the eruption of prominences into solar flares.     

Excitation of high-m tearing modes at the solar flare site

It is shown that high-m drift tearing modes can be excited under the conditions prevalent at the solar flare sites. Since the growth rate of the high-m tearing modes is larger than that for low-m macroscopic tearing modes and smaller than that of microscopic ion-acoustic instability, these modes warrant accommodation in the scheme of instabilities possibly operating in the hybrid model of solar flares suggested by Spicer.     

Observations of low-degree solar oscillations with few detector elements

The detection of low-degree solar oscillation modes with a specific low-resolution detector configuration is investigated. The detector is part of an instrument (the Luminosity Oscillations Imager) in the VIRGO package, to be flown on SOHO. Various problems such as p- and g-mode sensitivity, B and roll angle effects, modes isolation, cross-talk and guiding effects are treated for a given detector configuration. The computed sensitivity will enable the instrument to detect any type of modes for l < 6.B and roll angle effects can be compensated by using adequate filters for mode isolation. Guiding effects are small for p-modes. Also some other complex high-degree mode effects are treated.     

An Estimate of <Emphasis Type="Italic">P</Emphasis>-Mode Damping by Wave Leakage

High-cadence TRACE observations show that outward-propagating intensity disturbances are a common feature in large, quiescent coronal loops. Analysis of the frequency distribution of these modes shows peaks at both three- and five-minute periods, indicating that they may be driven by the solar surface oscillations (p modes). The energy flux contained within the coronal intensity disturbances is of the order of (1.1±0.4)×10³ ergs cm⁻² s⁻¹. A simple order-of-magnitude estimate of the damping rate of the relevant p modes allows us to put an observational constraint on the damping of p modes and shows that leakage into the overlying coronal atmosphere might be able to account for a significant fraction of p-mode damping.     

Global wave patterns in the Sun's magnetic field

When the observed pattern of solar magnetic fields is decomposed in its spherical-harmonic components and a time series analysis is performed, a resonant global wave pattern is revealed. The power spectrum indicates modes with discrete frequencies, obeying a strict parity selection rule in the case of the zonal, rotationally-symmetric modes (with spherical-harmonic orderm=0). For instance, the 22 yr resonance that dominates for the anti-symmetric modes (with odd values of the spherical-harmonic degreel) is completely absent for the symmetric modes, which instead exhibit a number of resonances having frequencies increasing withl.A more traditional way of looking at the evolution of the zonal magnetic pattern is in the form of isocontours in latitude-time space (as in the butterfly diagram of sunspots). We show how this pattern can to a good approximation be represented as a superposition of 14 discrete modes, each with a purely sinusoidal time variation, one mode for each value ofl (=1,2, ..., 14). This corresponds to the assumption that the true, fully resolved and noise-free power spectrum consists of -function peaks, one for eachl value.This approach allows us to analyse the roles of the individual discrete modes in generating the well-known features in the traditional btterfly diagrams, e.g., the drift of the sunspot zones towards the equator and the prominence zones towards the poles during the course of the 11 yr cycle. It is shown that these features are accounted for entirely by the odd parity modes with the single, sinusoidal period of 22 yr. The drifts (and thus the arrow of time) are caused by the systematic phase relations between the 22 yr modes. The even modes exhibit an entirely different pattern. Since they have considerably shorter periods, they cause an undulation of the odd-mode contour lines when superposed on the anti-symmetric pattern.The dispersion, amplitude, and phase relations of the discrete modes are given. It is indicated how they can be used in combination with spectral inversion techniques to determine the depth variation of the parameters in the governing global wave equation.Paper dedicated to Professor Hannes Alfvén on the occasion of his 80th birthday, 30 May 1988.     

Gravity Modes in Delta Scuti Stars

This paper examines the observational evidence for the detectionof gravity modes in Scuti stars, which are p-mode pulsators.Low-order gravity modes have also been found in at least one star (FG Vir).Some reports of gravity modes may be due to systematic errors in theabsolute magnitude calibrations for slowly rotating stars. Furthermore,many detected low frequencies are not high-order gravity modes, but linear combinations,f _i -f _j , of the main pulsation modes. Other low frequencies are caused bya close binary companion leading to tidal deformation as well as tidallyexcited gravity modes.     

Helical magnetorotational instability of Taylor‐Couette flows in the Rayleigh limit and for quasi‐Kepler rotation

The magnetorotational instability (MRI) of differential rotation under the simultaneous presence of axial and azimuthal components of the (current‐free) magnetic field is considered. For rotation with uniform specific angular momentum the MHD equations for axisymmetric perturbations are solved in a local short‐wave approximation. All the solutions are overstable for B_z · B_ϕ ≠ 0 with eigenfrequencies approaching the viscous frequency. For more flat rotation laws the results of the local approximation do not comply with the results of a global calculation of the MHD instability of Taylor‐Couette flows between rotating cylinders. – With B_ϕ and B_z of the same order the traveling‐mode solutions are also prefered for flat rotation laws such as the quasi‐Kepler rotation. For magnetic Prandtl number Pm → 0 they scale with the Reynolds number of rotation rather than with the magnetic Reynolds number (as for standard MRI) so that they can easily be realized in MHD laboratory experiments. – Regarding the nonaxisymmetric modes one finds a remarkable influence of the ratio B_ϕ/B_z only for the extrema. For B_ϕ ≫ B_z and for not too small Pm the nonaxisymmetric modes dominate the traveling axisymmetric modes. For standard MRI with B_z ≫ B_ϕ, however, the critical Reynolds numbers of the nonaxisymmetric modes exceed the values for the axisymmetric modes by many orders so that they are never prefered. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Mode Conversion of Solar p Modes in non-Vertical Magnetic Fields – i. two-Dimensional Model

Sunspots absorb incident p modes. The responsible mechanism is uncertain. One possibility is mode conversion to slow magnetoacoustic–gravity waves. In vertical field mode conversion can adequately explain the observed f-mode absorption, but is too inefficient to explain the absorption of p modes. In this investigation we calculate the efficiency of fast-to-slow magnetoacoustic–gravity wave conversion in non-vertical field. We assume two-dimensional propagation where the Alfvén waves decouple. It is found that resultant p-mode absorption is significantly enhanced for moderate inclinations at higher frequencies, whereas for p modes at lower frequencies, and the f mode in general, there is no useful enhancement. However, the enhancement is insufficient to explain the observed p-mode absorption by sunspots. Paper II considers the efficiency of mode conversion in non-vertical field with three-dimensional propagation, where fast and slow magnetoacoustic–gravity waves and Alfvén waves are coupled.     

Detection and temporal coherence of p‐modes below 1.4 mHz

Data collected recently by the helioseismic experiments aboard the SOHO spacecraft have allowed the detection of low degree p‐modes with increasingly lower order n. In particular, the GOLF experiment is currently able to unambiguously identify low degree modes with frequencies as low as 1.3 mHz. The detection of p‐modes with very low frequency (i.e., low n), is difficult due to the low signal‐to‐noise ratio in this spectral region and its contamination by solar signals that are not of acoustic origin. To address this problem without using any theoretical a priory, we propose a methodology that relies only on the inversion of observed values to define a spectral window for the expected locations of these low frequency modes. The application of this method to 2920‐day‐long GOLF observations is presented and its results discussed. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Modelling p-Mode Interaction with a Spreading Sunspot Field

Sunspots absorb and scatter incident p modes. The dominant mechanism is still uncertain. One possibility, mode conversion to slow magneto-acoustic waves, has been shown to yield results in agreement with observations for the f mode only. Absorption of p modes in simple vertical magnetic field models is too weak by an order of magnitude or more. Here we report on numerical calculations of p modes encountering a simple sunspot model with field which spreads with height. It is found that p-mode absorption is greatly enhanced by field spread, to a level consistent with observations, and it appears that it occurs preferentially in the outer regions of the spot, in line with recent results from acoustic holography.     

First phase heating and particle acceleration during solar flares by fast tearing modes

We develop a simple, but physically consistent, model of heating and particle acceleration by fast tearing modes, for modeling compact loop flares or erupting prominences. It is shown that there is a slow preheating, over many e -foldings of the instability, after which a rapid heating takes place in approximately one e-folding. The role of anomalous resistivity excited by the induced electric field during tearing is discussed, and how both thermal conduction and plasma expansion may play a role in cooling. Estimates for the total number of thermal and non-thermal electrons generated by one fast tearing mode are given, and it is argued that collisional tearing modes give rise to a primarily thermal plasma.     

Excitation of solar-like oscillations: From PMS to MS stellar models

The amplitude of solar-like oscillations results from a balance between excitation and damping. As in the sun, the excitation is attributed to turbulent motions that stochastically excite thep modes in the uppermost part of the convective zone. We present here a model for the excitation mechanism. Comparisons between modeled amplitudes and helio and stellar seismic constraints are presented and the discrepancies discussed. Finally the possibility and the interest of detecting such stochastically excited modes in pre-main sequence stars are also discussed.     

On fast magnetosonic coronal pulsations

The linear properties of the fast magnetosonic modes of a coronal loop modelled as a smooth density inhomogeneity in a uniform magnetic field are compared with the case of a step function slab. It is shown that the group velocityC _gof the modes, important in determining the structure of impulsively excited wave packets, possesses a minimum for a wide class of profile including the slab, with the exception of the Epstein profile for which the minimum inC _gmoves out to infinity. Results for the simple step profile are thus of wider validity, and likely to be applicable to coronal loops.     

The effect of turbulent mixing on the g‐mode spectrum of MS stars

Understanding transport processes inside stars is one of the main goals of asteroseismology. Chemical turbulent mixing can affect the internal distribution of μ near the energy generating core, having an effect on the evolutionary tracks similar to that of overshooting. This mixing leads to a smoother chemical composition profile near the edge of the convective core, which is reflected in the behavior of the buoyancy frequency and, therefore, in the frequencies of gravity modes. We describe the effects of convective overshooting and turbulent mixing on the frequencies of gravity modes in B‐type main sequence stars. In particular, the cases of p‐g mixed modes in β Cep stars and high‐order modes in SPBs are considered. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Mean-field dynamo with cubic non-linearity

The turbulent mean-field dynamo of αω-type with a mean helicity quadratically dependent on the magnetic field is investigated. A nonlinear system of ordinary differential equation is derived for the amplitudes of the magnetic field expansion over the eigenvectors of the linear problem. In a one-mode approximation the non-linear supercritical solution is stable when dγ/d D > 0, where γ is the growth rate of the linear solution and D is the dynamo number. Non-linear interation between two modes of dipole and quadrupole symmetry is considered. The conditions are found for the synchronization and beats of these modes under the assumption that the quadrupole mode is weaker than the dipole one.     

The equilibrium and stability of magnetopolytropes

The theory of the oscillations of axisymmetric gaseous configurations with a prevalent magnetic field is presented. The virial tensor method is used to obtain the nine second harmonic modes of oscillations of the system. It is found that out of the nine modes, three are neutral, four are non-radial, and two are coupled. For the Prendergast spherical model it is found that one of the coupled modes is radial and the other non-radial. Both the radial and the non-radial modes obtained in this case agree with the corresponding formulae obtained byChandrasekhar andLimber (1954) andWoltjer (1962).The equilibrium structure of gaseous polytropes with toroidal magnetic fields is also investigated in detail for values of the polytropic indexn=1, 1.5, 2, 3 and 3.5. For this model the components of the moment of intertia and potential energy tensors together with the non-zero components of the supermatrix potential are obtained. The final results in terms of the effect of weak toroidal magnetic fields on the characteristic frequencies of distorted polytropes are presented in the form of tables.     

Small-Amplitude Disturbances In A Radiating And Scattering Grey Medium Ii. Solutions Of Given Real Wave Number <Emphasis Type="BoldItalic">k</Emphasis>

We study the fundamental modes of radiation hydrodynamic waves arising from one-dimensional small-amplitude initial fluctuations with wave number k in a radiating and scattering grey medium using the Eddington approximation. The dispersion relation analyzed is the same as that of Paper I (Kaneko et al., 2000), but is solved as a quintic in angular frequency ω while a quadratic in k ² in Paper I. Numerical results reveal that wave patterns of five solutions are distinguished into three types of the radiation-dominated and type 1 and type 2 matter-dominated cases. The following wave modes appear in our problem: radiation wave, conservative radiation wave, entropy wave, Newtonian-cooling wave, opacity-damped and cooling-damped waves, constant-volume and constant-pressure diffusion modes, adiabatic sound wave, cooling-damped and drag–force-damped isothermal sound waves, isentropic radiation-acoustic wave, and gap mode. The radiation-dominated case is characterized by the gap between the isothermal sound and isentropic radiation-acoustic speeds within which there is not any acoustic wave propagating with real phase speed. One of the differences between type 1 and type 2 matter-dominated cases is the connectivity of the constant-volume diffusion mode, which originates from the radiative mode in the former case, while from the Newtonian-cooling wave in the latter case. Analytic solutions are derived for all wave modes to discuss their physical significance. The criterion, which distinguishes between radiation-dominated and type 1 matter-dominated cases, is given by Γ₀ = 9, where Γ₀ = C _p ^(tot)/C _V ^(tot) is the ratio of total specific heats at constant pressure and constant volume. Waves in a scattering grey medium are also analyzed, which provides us some hints for the effects of energy and momentum exchange between matter and radiation.