Theoretical and Experimental Studies of the Rigidity Spectrum of the 27-Day Variation of the Galactic Cosmic Ray Intensity in Different Epochs of Solar Activity

Recently Pop (Solar Phys. 276, 351, 2012) identified a Laplace (or double exponential) distribution in the number of days with a given absolute value in the change over a day, in sunspot number, for days on which the sunspot number does change. We show this phenomenological rule has a physical origin attributable to sunspot formation, evolution, and decay, rather than being due to the changes in sunspot number caused by groups rotating onto and off the visible disc. We also demonstrate a simple method to simulate daily sunspot numbers over a solar cycle using the Pop (Solar Phys. 276, 351, 2012) result, together with a model for the cycle variation in the mean sunspot number. The procedure is applied to three recent solar cycles. We check that the simulated sunspot numbers reproduce the observed distribution of daily changes over those cycles.     

Estimating Flare-Related Photospheric Lorentz Force Vector Changes Within Active Regions

It is shown that expressions for the global Lorentz force associated with a flaring active region derived by Fisher et al. (Solar Phys. 277, 59, 2012) can be used to estimate the Lorentz-force changes for strong fields in large structures over photospheric subdomains within active regions. Gary’s (Solar Phys. 203, 71, 2001) model for the stratified solar atmosphere is used to demonstrate that in large-scale structures with typical horizontal magnetic length scale ??300 km and with strong magnetic fields (≥?1 kG at the τ=1 opacity layer at 5000 Å), the Lorentz force acting on the photosphere may be approximated by a surface integral based on photospheric boundary data alone. These conditions cover many of the sunspot fields and major neutral lines that have been studied using Fisher et al.’s (2012) expressions over the past few years. The method gives a reasonable estimate of flare-related Lorentz-force changes based on photospheric magnetogram observations provided that the Lorentz-force changes associated with the flare have a lasting effect on the observed fields, and they are not immediately erased by post-flare equilibration processes.     

Very Fast and Accurate Azimuth Disambiguation of Vector Magnetograms

We present a method for fast and accurate azimuth disambiguation of vector magnetogram data regardless of the location of the analyzed region on the solar disk. The direction of the transverse field is determined with the principle of minimum deviation of the field from the reference (potential) field. The new disambiguation (NDA) code is examined on the well-known models of Metcalf et al. (Solar Phys. 237, 267, 2006) and Leka et al. (Solar Phys. 260, 83, 2009), and on an artificial model based on the observed magnetic field of AR 10930 (Rudenko, Myshyakov, and Anfinogentov, Astron. Rep. 57, 622, 2013). We compare Hinode/SOT-SP vector magnetograms of AR 10930 disambiguated with three codes: the NDA code, the nonpotential magnetic-field calculation (NPFC: Georgoulis, Astrophys. J. Lett. 629, L69, 2005), and the spherical minimum-energy method (Rudenko, Myshyakov, and Anfinogentov, Astron. Rep. 57, 622, 2013). We then illustrate the performance of NDA on SDO/HMI full-disk magnetic-field observations. We show that our new algorithm is more than four times faster than the fastest algorithm that provides the disambiguation with a satisfactory accuracy (NPFC). At the same time, its accuracy is similar to that of the minimum-energy method (a very slow algorithm). In contrast to other codes, the NDA code maintains high accuracy when the region to be analyzed is very close to the limb.     

Implementation and Comparison of Acoustic Travel-Time Measurement Procedures for the Solar Dynamics Observatory/Helioseismic and Magnetic Imager Time?–?Distance Helioseismology Pipeline

The Helioseismic and Magnetic Imager (HMI) instrument onboard the Solar Dynamics Observatory (SDO) satellite is designed to produce high-resolution Doppler-velocity maps of oscillations at the solar surface with high temporal cadence. To take advantage of these high-quality oscillation data, a?time?–?distance helioseismology pipeline (Zhao et al., Solar Phys. submitted, 2010) has been implemented at the Joint Science Operations Center (JSOC) at Stanford University. The aim of this pipeline is to generate maps of acoustic travel times from oscillations on the solar surface, and to infer subsurface 3D flow velocities and sound-speed perturbations. The wave travel times are measured from cross-covariances of the observed solar oscillation signals. For implementation into the pipeline we have investigated three different travel-time definitions developed in time?–?distance helioseismology: a Gabor-wavelet fitting (Kosovichev and Duvall, SCORE’96: Solar Convection and Oscillations and Their Relationship, ASSL, Dordrecht, 241, 1997), a?minimization relative to a reference cross-covariance function (Gizon and Birch, Astrophys. J. 571, 966, 2002), and a linearized version of the minimization method (Gizon and Birch, Astrophys. J. 614, 472, 2004). Using Doppler-velocity data from the Michelson Doppler Imager (MDI) instrument onboard SOHO, we tested and compared these definitions for the mean and difference travel-time perturbations measured from reciprocal signals. Although all three procedures return similar travel times in a quiet-Sun region, the method of Gizon and Birch (Astrophys. J. 614, 472, 2004) gives travel times that are significantly different from the others in a magnetic (active) region. Thus, for the pipeline implementation we chose the procedures of Kosovichev and Duvall (SCORE’96: Solar Convection and Oscillations and Their Relationship, ASSL, Dordrecht, 241, 1997) and Gizon and Birch (Astrophys. J. 571, 966, 2002). We investigated the relationships among these three travel-time definitions, their sensitivities to fitting parameters, and estimated the random errors that they produce.     

Preprocessing the Photospheric Vector Magnetograms for an NLFFF Extrapolation Using a Potential-Field Model and an Optimization Method

Numerical reconstruction/extrapolation of the coronal nonlinear force-free magnetic field (NLFFF) usually takes the photospheric vector magnetogram as input at the bottom boundary. The magnetic field observed at the photosphere, however, contains a force that is in conflict with the fundamental assumption of the force-free model. It also contains measurement noise, which hinders the practical computation. Wiegelmann, Inhester, and Sakurai (Solar Phys. 233, 215, 2006) have proposed to preprocess the raw magnetogram to remove the force and noise to provide better input for NLFFF modeling. In this paper we develop a new code of magnetogram preprocessing that is consistent with our extrapolation method CESE–MHD–NLFFF (Jiang, Feng, and Xiang in Astrophys. J. 755, 62, 2012; Jiang and Feng in Astrophys. J. 749, 135, 2012a). Based on the magnetic-splitting rule that a magnetic field can be split into a potential-field part and a non-potential part, we split the magnetogram and dealt with the two parts separately. The preprocessing of the magnetogram’s potential part is based on a numerical potential-field model, and the non-potential part is preprocessed using the similar optimization method of Wiegelmann, Inhester, and Sakurai (2006). The code was applied to the SDO/HMI data, and results show that the method can remove the force and noise efficiently and improve the extrapolation quality.     

Primordial magnetic fields constrained from CMB anisotropies on dynamo cosmology

Magneto-curvature stresses could deform magnetic field lines giving rise to back reaction and restoring magnetic stresses (Tsagas in Phys. Rev. Lett., 2001). Barrow and Tsagas (Phys. Rev. D, 2008) have shown that in Friedman universe the expansion slows down in its spatial section of negative Riemann curvature. Earlier, Chicone and Latushkin (Proc. Am. Math. Soc. 125(11):3391, 1995) proved that fast dynamos in compact 2D manifold implies negatively constant Riemannian curvature. Here one applies the Barrow-Tsagas ideas to cosmic dynamos of negative curvature. Fast dynamo, covariant stretching of Riemann slices of cosmic Lobachevsky plane is given. Inclusion of advection term on dynamo equations (Clarkson and Marklund in Mon. Not. R. Astron. Soc., 2005) is considered. In advection absence, slow dynamos are also obtained. It is shown the viscous and restoring forces on stretching particles decrease, as magnetic rates increase. From COBE data ( $\frac{{\delta}B}{B}\approx{10^{-5}}$ ), one is able to compute the stretching $\frac{{\delta}V^{y}}{V^{y}}=1.5\frac{{\delta}B}{B}\approx{1.5{\times}10^{-5}}$ . Zeldovich et al. have computed the maximum magnetic growth rate as γ _max ≈8.0×10^?1 t ^?1. From COBE data a lower growth rate as γ _COBE ≈6.0×10^?6 t ^?1, is well-within Zeldovich et al estimate. Instead of Harrison value $B\approx{t^{\frac{4}{3}}}$ one obtains a lower primordial field B≈10^?6 t which yields B≈10^?6 G at 1 s Big Bang time.     

Validation of the 3D AMR SIP–CESE Solar Wind Model for Four Carrington Rotations

We carry out the adaptive mesh refinement (AMR) implementation of our solar–interplanetary space-time conservation element and solution element (CESE) magnetohydrodynamic model (SIP–CESE MHD model) using a six-component grid system (Feng, Zhou, and Wu, Astrophys. J. 655, 1110, 2007; Feng et al., Astrophys. J. 723, 300, 2010). By transforming the governing MHD equations from the physical space (x,y,z) to the computational space (ξ,η,ζ) while retaining the form of conservation (Jiang et al., Solar Phys. 267, 463, 2010), the SIP–AMR–CESE MHD model is implemented in the reference coordinates with the aid of the parallel AMR package PARAMESH available at http://sourceforge.net/projects/paramesh/ . Meanwhile, the volumetric heating source terms derived from the topology of the magnetic-field expansion factor and the minimum angular separation (at the photosphere) between an open-field foot point and its nearest coronal-hole boundary are also included. We show the preliminary results of applying the SIP–AMR–CESE MHD model for simulating the solar-wind background of different solar-activity phases by comparison with SOHO observations and other spacecraft data from OMNI. Our numerical results show overall good agreements in the solar corona and in interplanetary space with these multiple-spacecraft observations.     

Changes in Quasi-periodic Variations of Solar Photospheric Fields: Precursor to the Deep Solar Minimum in Cycle 23?

Possible precursor signatures in the quasi-periodic variations of solar photospheric fields were investigated in the build-up to one of the deepest solar minima experienced in the past 100 years. This unusual and deep solar minimum occurred between Solar Cycles 23 and 24. We used both wavelet and Fourier analysis to study the changes in the quasi-periodic variations of solar photospheric fields. Photospheric fields were derived using ground-based synoptic magnetograms spanning the period 1975.14 to 2009.86 and covering Solar Cycles 21, 22, and 23. A hemispheric asymmetry in the periodicities of the photospheric fields was seen only at latitudes above ±?45^° when the data were divided into two parts based on a wavelet analysis: one prior to 1996 and the other after 1996. Furthermore, the hemispheric asymmetry was observed to be confined to the latitude range of 45^° to 60^°. This can be attributed to the variations in polar surges that primarily depend on both the emergence of surface magnetic flux and varying solar-surface flows. The observed asymmetry along with the fact that both solar fields above ±?45^° and micro-turbulence levels in the inner-heliosphere have been decreasing since the early- to mid-nineties (Janardhan et al. in Geophys. Res. Lett. 382, 20108, 2011) suggest that around this time active changes occurred in the solar dynamo that governs the underlying basic processes in the Sun. These changes in turn probably initiated the build-up to the very deep solar minimum at the end of Cycle 23. The decline in fields above ±?45^°, for well over a solar cycle, would imply that weak polar fields have been generated in the past two successive solar cycles, viz. Cycles 22 and 23. A continuation of this declining trend beyond 22 years, if it occurs, will have serious implications for our current understanding of the solar dynamo.     

Quantitative Analysis of CME Deflections in the Corona

In this paper, ten CME events viewed by the STEREO twin spacecraft are analyzed to study the deflections of CMEs during their propagation in the corona. Based on the three-dimensional information of the CMEs derived by the graduated cylindrical shell (GCS) model (Thernisien, Howard, and Vourlidas in Astrophys. J. 652, 1305, 2006), it is found that the propagation directions of eight CMEs had changed. By applying the theoretical method proposed by Shen et?al. (Solar Phys. 269, 389, 2011) to all the CMEs, we found that the deflections are consistent, in strength and direction, with the gradient of the magnetic energy density. There is a positive correlation between the deflection rate and the strength of the magnetic energy density gradient and a weak anti-correlation between the deflection rate and the CME speed. Our results suggest that the deflections of CMEs are mainly controlled by the background magnetic field and can be quantitatively described by the magnetic energy density gradient (MEDG) model.     

Properties of the 15 February 2011 Flare Seismic Sources

The first near-side X-class flare of Solar Cycle 24 occurred in February 2011 (SOL2011-02-05T01:55) and produced a very strong seismic response in the photosphere. One sunquake was reported by Kosovichev (Astrophys. J. Lett. 734, L15, 2011), followed by the discovery of a second sunquake by Zharkov, Green, Matthews et al. (Astrophys. J. Lett. 741, L35, 2011). The flare had a two-ribbon structure and was associated with a flux-rope eruption and a halo coronal mass ejection (CME) as reported in the CACTus catalogue. Following the discovery of the second sunquake and the spatial association of both sources with the locations of the feet of the erupting flux rope (Zharkov, Green, Matthews et al., Astrophys. J. Lett. 741, L35, 2011), we present here a more detailed analysis of the observed photospheric changes in and around the seismic sources. These sunquakes are quite unusual, taking place early in the impulsive stage of the flare, with the seismic sources showing little hard X-ray (HXR) emission, and strongest X-ray emission sources located in the flare ribbons. We present a directional time–distance diagram computed for the second source, which clearly shows a ridge corresponding to the travelling acoustic-wave packet and find that the sunquake at the second source happened about 45 seconds to one minute earlier than the first source. Using acoustic holography we report different frequency responses of the two sources. We find strong downflows at both seismic locations and a supersonic horizontal motion at the second site of acoustic-wave excitation.     

Capture probability in the 3:1 mean motion resonance with Jupiter: an application to the Vesta family

We study the capture and crossing probabilities in the 3:1 mean motion resonance with Jupiter for a small asteroid that migrates from the inner to the middle Main Belt under the action of the Yarkovsky effect. We use an algebraic mapping of the averaged planar restricted three-body problem based on the symplectic mapping of Hadjidemetriou (Celest Mech Dyn Astron 56:563–599, 1993), adding the secular variations of the orbit of Jupiter and non-symplectic terms to simulate the migration. We found that, for fast migration rates, the captures occur at discrete windows of initial eccentricities whose specific locations depend on the initial resonant angles, indicating that the capture phenomenon is not probabilistic. For slow migration rates, these windows become narrower and start to accumulate at low eccentricities, generating a region of mutual overlap where the capture probability tends to 100 %, in agreement with the theoretical predictions for the adiabatic regime. Our simulations allow us to predict the capture probabilities in both the adiabatic and non-adiabatic cases, in good agreement with results of Gomes (Celest Mech Dyn Astron 61:97–113, 1995) and Quillen (Mon Not RAS 365:1367–1382, 2006). We apply our model to the case of the Vesta asteroid family in the same context as Roig et al. (Icarus 194:125–136, 2008), and found results indicating that the high capture probability of Vesta family members into the 3:1 mean motion resonance is basically governed by the eccentricity of Jupiter and its secular variations.     

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).     

Perturbations of a geo-centric synchronous satellite with resonance

We have investigated the resonances due to the perturbations of a geo-centric synchronous satellite under the gravitational forces of the Sun, the Moon and the Earth including it’s equatorial ellipticity. The resonances at the points resulting from (i) the commensurability between \(\dot{\theta}_{0}\) (steady-state orbital angular rate of the satellite) and \(\dot{\theta}_{m}\) (angular velocity of the moon around the earth) and (ii) the commensurability between \(\dot{\theta}_{0}\) and \(\dot{\psi}_{0}\) (steady-state regression rate of the synchronous satellite) are analyzed. The amplitude and the time period of the oscillation have been determined by using the procedure as given in Brown and Shook (Planetary Theory, Cambridge University Press, Cambridge, 1933). We have observed that as θ _m (0^°≤θ _m ≤45^°) and ψ (0^°≤ψ≤135^°) increase, the amplitude decreases and the time period also decreases. We have also shown the effect of ψ on amplitude and time period for 0^°≤Γ≤45^°, where Γ is the angle measured from the minor axis of the earth’s equatorial ellipse to the projection of the satellite on the plane of the equator.     

Forecasting the Maxima of Solar Cycle 24 with Coronal Fe xiv Emission

The onset of the “Rush to the Poles” of polar-crown prominences and their associated coronal emission is a harbinger of solar maximum. Altrock (Solar Phys. 216, 343, 2003) showed that the “Rush” was well observed at 1.15 R _o in the Fe xiv corona at the Sacramento Peak site of the National Solar Observatory prior to the maxima of Cycles 21 to 23. The data show that solar maximum in those cycles occurred when the center line of the Rush reached a critical latitude of 76^°±2^°. Furthermore, in the previous three cycles solar maximum occurred when the highest number of Fe xiv emission features per day (averaged over 365 days and both hemispheres) first reached latitudes 20^°±1.7^°. Applying the above conclusions to Cycle 24 is difficult due to the unusual nature of this cycle. Cycle 24 displays an intermittent Rush that is only well-defined in the northern hemisphere. In 2009 an initial slope of 4.6^°?year^?1 was found in the north, compared to an average of 9.4±1.7^°?year^?1 in the previous cycles. An early fit to the Rush would have reached 76^° at 2014.6. However, in 2010 the slope increased to 7.5^°?year^?1 (an increase did not occur in the previous three cycles). Extending that rate to 76^°±2^° indicates that the solar maximum in the northern hemisphere already occurred at 2011.6±0.3. In the southern hemisphere the Rush to the Poles, if it exists, is very poorly defined. A linear fit to several maxima would reach 76^° in the south at 2014.2. In 1999, persistent Fe xiv coronal emission known as the “extended solar cycle” appeared near 70^° in the North and began migrating towards the equator at a rate 40 % slower than the previous two solar cycles. However, in 2009 and 2010 an acceleration occurred. Currently the greatest number of emission features is at 21^° in the North and 24^° in the South. This indicates that solar maximum is occurring now in the North but not yet in the South.     

Higher dimensional exact solutions for a charged fluid sphere in general theory of relativity

The exact higher dimensional solutions of Einstein-Maxwell field equations for spherically symmetric distribution of charged perfect fluid are obtained by using the method originally used by Hajj-Boutros and Sfeila (Gen. Relativ. Gravit. 18(4):395, 1986) for four-dimensional space-time. The new exact solutions have been generated from those of Khadekar et al. (J. Indian Math. Soc. 68(1–4):33, 2001), Humi and Mansour (Phys. Rev. D 29(6):1076, 1984) and Banerjee and Santos (J. Math. Phys. 22(4):824, 1981) in the frame work of higher dimensional space-time. The various physical properties are also discussed.     

Magneto-Acoustic Energetics Study of the Seismically Active Flare of 15 February 2011

Multi-wavelength studies of energetic solar flares with seismic emissions have revealed interesting common features between them. We studied the first GOES X-class flare of Solar Cycle 24, as detected by the Solar Dynamics Observatory (SDO). For context, seismic activity from this flare (SOL2011-02-15T01:55-X2.2, in NOAA AR 11158) has been reported by Kosovichev (Astrophys. J. Lett., 734, L15, 2011) and Zharkov et?al. (Astrophys. J. Lett., 741, L35, 2011). Based on Dopplergram data from the Helioseismic and Magnetic Imager (HMI), we applied standard methods of local helioseismology in order to identify the seismic sources in this event. RHESSI hard X-ray data are used to check the correlation between the location of the seismic sources and the particle-precipitation sites in during the flare. Using HMI magnetogram data, the temporal profile of fluctuations in the photospheric line-of-sight magnetic field is used to estimate the magnetic-field change in the region where the seismic signal was observed. This leads to an estimate of the work done by the Lorentz-force transient on the photosphere of the source region. In this instance, this is found to be a significant fraction of the acoustic energy in the attendant seismic emission, suggesting that Lorentz forces can contribute significantly to the generation of sunquakes. However, there are regions in which the signature of the Lorentz force is much stronger, but from which no significant acoustic emission emanates.     

A Statistical Study on CMEs Associated with DH-Type-II Radio Bursts Based on Their Source Location (Limb and Disk Events)

We have statistically studied the 344 Coronal Mass Ejections (CMEs) associated with flares and DH-type-II radio bursts (1??C?14 MHz) during 1997??C?2008. We found that only 3?% of the total CMEs (344) compared to the general population CMEs (13208) drives DH-type-II radio bursts (Gopalswamy in Solar Eruptions and Energetic Particles, AGU Geophys. Monogr. 165, 207, 2006). Out of 344 events we have selected 236 events for further analysis. We divided the events into two groups: i) disk events (within 45° from the disk center) and ii) limb events (beyond 45° but within 90° from the disk center). We find that the average CME speed of the limb events (1370?km?s^?1) is three times, while for the disk events (1055?km?s^?1) it is two times the average speed of the general population CMEs (433?km?s^?1). The average widths of the limb events (129°) and disk events (116°) are two times greater than the average width of the general population CMEs (58°). We found a better correlation between the CME speed and width (correlation coefficient R=0.56) for the limb events than that of the disk events (R=0.47). The shock speed of the CMEs associated with DH-type-II radio bursts is found by applying Leblanc, Dulk, and Bougeret??s (Solar Phys. 183, 165, 1998) electron density model; the disk events are found to have an average speed of 1190 km?s^?1 and that of the limb events is 1275 km?s^?1. From this study we compare the CME properties between limb and disk events. The properties like CME speed, width, shock speed, and correlation between CME speed and width are found to be higher for limb events than disk events. The results in disk events are subject to projection effects, and this study stresses the importance of these effects.     

Transient Artifacts in a Flare Observed by the Helioseismic and Magnetic Imager on the Solar Dynamics Observatory

The Helioseismic and Magnetic Imager (HMI) onboard the Solar Dynamics Observatory (SDO) provides a new tool for the systematic observation of white-light flares, including Doppler and magnetic information as well as continuum. In our initial analysis of the highly impulsive $\mathrm{\gamma}$ -ray flare SOL2010-06-12T00:57 (Martínez Oliveros et al., Solar Phys. 269, 269, 2011), we reported the signature of a strong blueshift in the two footpoint sources. Concerned that this might be an artifact due to aliasing peculiar to the HMI instrument, we undertook a comparative analysis of Global Oscillation Network Group (GONG++) observations of the same flare, using the PArametric Smearing Correction ALgorithm (PASCAL) algorithm to correct for artifacts caused by variations in atmospheric smearing. This analysis confirms the artifactual nature of the apparent blueshift in the HMI observations, finding weak redshifts at the footpoints instead. We describe the use of PASCAL with GONG++ observations as a complement to the SDO observations and discuss constraints imposed by the use of HMI far from its design conditions. With proper precautions, these data provide rich information on flares and transients.     

Erratum to: Reports of the IAU Working Group on Cartographic Coordinates and Rotational Elements: 2006 & 2009

The primary poles for (243) Ida and (134340) Pluto and its satellite (134340) Pluto : I Charon were redefined in the IAU Working Group on Cartographic Coordinates and Rotational Elements (WGCCRE) 2006 report (Seidelmann et al. in Celest Mech Dyn Astr 98:155, 2007), and 2009 report (Archinal et al. in Celest Mech Dyn Astr 109:101, 2011), respectively, to be consistent with the primary poles of similar Solar System bodies. However, the WGCCRE failed to take into account the effect of the redefinition of the poles on the values of the rotation angle W at J2000.0. The revised relationships in Table 3 of Archinal et al. 2011) are $$\begin{array}{llll} W & = & 274^{\circ}.05 +1864^{\circ}.6280070\, d\;{\rm for\; (243)\,Ida} \\ W & = & 302^{\circ} .695 + 56^{\circ} .3625225\, d\;{\rm for\; (134340)\,Pluto,\; and}\\ W & = & 122^{\circ} .695 + 56^{\circ} .3625225\, d\;{\rm for\; (134340)\,Pluto : I \,Charon}\end{array}$$ where d is the time in TDB days from J2000.0 (JD2451545.0).     

Effects of Random Flows on the Solar f Mode: II. Horizontal and Vertical Flow

We study the influence of horizontal and vertical random flows on the solar f mode in a plane-parallel, incompressible model that includes a static atmosphere. The incompressible limit is an adequate approximation for f-mode type of surface waves that are highly incompressible. The paper revisits and extends the problem investigated earlier by Murawski and Roberts (Astron. Astrophys. 272, 601, 1993). We show that the consideration of the proposed velocity profile requires several restrictive assumptions to be made. These constraints were not recognised in previous studies. The impact of the inconsistencies in earlier modelling is analysed in detail. Corrections to the dispersion relation are derived and the relevance of these corrections is analysed. Finally, the importance of the obtained results is investigated in the context of recent helioseismological data. Detailed comparison with our complementary studies on random horizontal flows (Mole, Kerekes, and Erdélyi, Solar Phys., accepted, 2008) and the random magnetic model of Erdélyi, Kerekes, and Mole (Astron. Astrophys. 431, 1083, 2005) is also given. In particular, for realistic solar parameters we find significant frequency reduction and wave damping, both of which increase with the characteristic thickness of the random layer.