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Register a new userInference of solar rotation from perturbations of acoustic mode eigenfunctions
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Todays picture of the internal solar rotation rate profile results essentially from helioseismic analyses of frequency splittings of resonant acoustic waves. Here we present another, complementary estimation of the internal solar rotation rate using the perturbation of the shape of the acoustic waves. For this purpose, we extend a global helioseismic approach developed previously for the investigation of the meridional flow cite{schad11, schad12, schad13} to work on the components of the differential rotation. We discuss the effect of rotation on mode eigenfunctions and thereon based observables. Based on a numerical study using a simulated rotation rate profile we tailor an inversion approach and also consider the case of the presence of an additional meridional flow. This inversion approach is then applied to data from the MDI (Michelson Doppler Imager aboard the Solar Heliospheric Observatory (SoHO)) instrument and the HMI (Helioseismic and Magnetic Imager aboard the Solar Dynamics Observatory (SDO)) instrument. In the end, rotation rate profiles estimated from eigenfunction perturbation and frequency splittings are compared. The rotation rate profiles from the two different approaches are qualitatively in good agreement, especially for the MDI data. Significant differences are obtained at high latitudes $> 50^{circ}$ and near the subsurface. The result from HMI data shows larger discrepancies between the different methods. We find that the two global helioseismic approaches provide complementary methods for measuring solar rotation. Comparing the results from different methods may help to reveal systematic influences that affect analyses based on eigenfunction perturbations, like meridional flow measurements.
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Helioseismology is the study of the solar interior using observations of oscillations at the surface. It suffers from systematic errors, such as a center-to-limb error in travel-time measurements. Understanding these errors requires a good understanding of the nontrivial relationship between wave displacement and helioseismic observables. The wave displacement causes perturbations in the atmospheric thermodynamical quantities which perturb the opacity, the optical depth, the source function, and the local ray geometry, thus affecting the emergent intensity. We aim to establish the most complete relationship up to now between the displacement and the intensity perturbation by solving the radiative transfer problem in the atmosphere. We derive an expression for the intensity perturbation caused by acoustic oscillations at any point on the solar disk by applying the first-order perturbation theory. As input, we consider adiabatic modes of oscillation of different degrees. The background and the perturbed intensities are computed considering the main sources of opacity in the continuum. We find that, for all modes, the perturbations to the thermodynamical quantities are not sufficient to model the intensity. In addition, the geometrical effects due to the displacement must be taken into account as they lead to a difference in amplitude and a phase shift between the temperature at the surface and intensity perturbations. The closer to the limb, the larger the differences. This work presents improvements for the computation of the intensity perturbations, in particular for high-degree modes, and explains differences in intensity computations in earlier works. The phase shifts and amplitude differences between the temperature and intensity perturbations increase towards the limb. This should help to interpret some of the systematic center-to-limb effects observed in local helioseismology.
The solar gravitational moments $J_{2n}$ are important astronomical quantities whose precise determination is relevant for solar physics, gravitational theory and high precision astrometry and celestial mechanics. Accordingly, we propose in the present work to calculate new values of $J_{2n}$ (for $n$=1,2,3,4 and 5) using recent two-dimensional rotation rates inferred from the high resolution SDO/HMI helioseismic data spanning the whole solar activity cycle 24. To this aim, a general integral equation relating $J_{2n}$ to the solar internal density and rotation is derived from the structure equations governing the equilibrium of slowly rotating stars. For comparison purpose, the calculations are also performed using rotation rates obtained from a recently improved analysis of SoHO/MDI heliseismic data for solar cycle 23. In agreement with earlier findings, the results confirmed the sensitivity of high order moments ($n>1$) to the radial and latitudinal distribution of rotation in the convective zone. The computed value of the quadrupole moment $J_{2}$ ($n=1$) is in accordance with recent measurements of the precession of Mercurys perihelion deduced from high precision ranging data of the MESSENGER spacecraft. The theoretical estimate of the related solar oblateness $Delta_{odot}$ is consistent with the most accurate space-based determinations, particularly the one from RHESSI/SAS.
We investigate the spatiotemporal structure of simulations of the homogeneous slab and isothermal plane models for the vertical motion in the Galactic disc. We use Dynamic Mode Decomposition (DMD) to compute eigenfunctions of the simulated distribution functions for both models, referred to as DMD modes. In the case of the homogeneous slab, we compare the DMD modes to the analytic normal modes of the system to evaluate the feasibility of DMD in collisionless self gravitating systems. This is followed by the isothermal plane model, where we focus on the effect of self gravity on phase mixing. We compute DMD modes of the system for varying relative dominance of mutual interaction and external potential, so as to study the corresponding variance in mode structure and lifetime. We find that there is a regime of relative dominance, at approximately $ 4:1 $ external potential to mutual interaction where the DMD modes are spirals in the $ (z,v_z) $ plane, and are nearly un-damped. This leads to the proposition that a system undergoing phase mixing in the presence of weak to moderate self gravity can have persisting spiral structure in the form of such modes. We then conclude with the conjecture that such a mechanism may be at work in the phase space spirals observed in Gaia Data Release 2, and that studying more complex simulations with DMD may aid in understanding both the timing and form of the perturbation that lead to the observed spirals.
We report analysis of sub-Alfvenic magnetohydrodynamic (MHD) perturbations in the low-b{eta} radial-field solar wind using the Parker Solar Probe spacecraft data from 31 October to 12 November 2018. We calculate wave vectors using the singular value decomposition method and separate the MHD perturbations into three types of linear eigenmodes (Alfven, fast, and slow modes) to explore the properties of the sub-Alfvenic perturbations and the role of compressible perturbations in solar wind heating. The MHD perturbations there show a high degree of Alfvenicity in the radial-field solar wind, with the energy fraction of Alfven modes dominating (~45%-83%) over those of fast modes (~16%-43%) and slow modes (~1%-19%). We present a detailed analysis of a representative event on 10 November 2018. Observations show that fast modes dominate magnetic compressibility, whereas slow modes dominate density compressibility. The energy damping rate of compressible modes is comparable to the heating rate, suggesting the collisionless damping of compressible modes could be significant for solar wind heating. These results are valuable for further studies of the imbalanced turbulence near the Sun and possible heating effects of compressible modes at MHD scales in low-b{eta} plasma.
Retrograde-propagating waves of vertical vorticity with longitudinal wavenumbers between 3 and 15 have been observed on the Sun with a dispersion relation close to that of classical sectoral Rossby waves. The observed vorticity eigenfunctions are symmetric in latitude, peak at the equator, switch sign near $20^circ$-$30^circ$, and decrease at higher latitudes. We search for an explanation that takes into account solar latitudinal differential rotation. In the equatorial $beta$ plane, we study the propagation of linear Rossby waves (phase speed $c <0$) in a parabolic zonal shear flow, $U = - overline{U} xi^2<0$, where $overline{U} = 244$ m/s and $xi$ is the sine of latitude. In the inviscid case, the eigenvalue spectrum is real and continuous and the velocity stream functions are singular at the critical latitudes where $U = c$. We add eddy viscosity in the problem to account for wave attenuation. In the viscous case, the stream functions are solution of a fourth-order modified Orr-Sommerfeld equation. Eigenvalues are complex and discrete. For reasonable values of the eddy viscosity corresponding to supergranular scales and above (Reynolds number $100 le Re le 700$), all modes are stable. At fixed longitudinal wavenumber, the least damped mode is a symmetric mode with a real frequency close to that of the classical Rossby mode, which we call the R mode. For $Re approx 300$, the attenuation and the real part of the eigenfunction is in qualitative agreement with the observations (unlike the imaginary part of the eigenfunction, which has a larger amplitude in the model. Conclusion: Each longitudinal wavenumber is associated with a latitudinally symmetric R mode trapped at low latitudes by solar differential rotation. In the viscous model, R modes transport significant angular momentum from the dissipation layers towards the equator.
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