Hankel Analysis Research Articles

A new look at sunspot formation using theory and observations

AbstractSunspots are of basic interest in the study of the Sun. Their relevance ranges from them being an activity indicator of magnetic fields to being the place where coronal mass ejections and flares erupt. They are therefore also an important ingredient of space weather. Their formation, however, is still an unresolved problem in solar physics. Observations utilize just 2D surface information near the spot, but it is debatable how to infer deep structures and properties from local helioseismology. For a long time, it was believed that flux tubes rising from the bottom of the convection zone are the origin of the bipolar sunspot structure seen on the solar surface. However, this theory has been challenged, in particular recently by new surface observation, helioseismic inversions, and numerical models of convective dynamos. In this article we discuss another theoretical approach to the formation of sunspots: the negative effective magnetic pressure instability. This is a large-scale instability, in which the total (kinetic plus magnetic) turbulent pressure can be suppressed in the presence of a weak large-scale magnetic field, leading to a converging downflow, which eventually concentrates the magnetic field within it. Numerical simulations of forced stratified turbulence have been able to produce strong super-equipartition flux concentrations, similar to sunspots at the solar surface. In this framework, sunspots would only form close to the surface due to the instability constraints on stratification and rotation. Additionally, we present some ideas from local helioseismology, where we plan to use the Hankel analysis to study the pre-emergence phase of a sunspot and to constrain its deep structure and formation mechanism.

THE SCATTERING OFf- ANDp-MODES FROM ENSEMBLES OF THIN MAGNETIC FLUX TUBES: AN ANALYTICAL APPROACH

Motivated by the observational results of Braun (1995), we extend the model of Hanson & Cally (2014) to address the effect of multiple scattering of f and p-modes by an ensemble of thin vertical magnetic flux tubes in the surface layers of the Sun. As in observational Hankel analysis we measure the scatter and phase shift from an incident cylindrical wave in a coordinate system roughly centred in the core of the ensemble. It is demonstrated that, although thin flux tubes are unable to interact with high order fluting modes individually, they can indirectly absorb energy from these waves through the scatters of kink and sausage components. It is also shown how the distribution of absorption and phase shift across the azimuthal order m depends strongly on the tube position, as well as on the individual tube characteristics. This is the first analytical study into an ensembles multiple scattering regime, that is embedded within a stratified atmosphere.

Oscillation Power in Sunspots and Quiet Sun from Hankel Analysis Performed on SDO/HMI and SDO/AIA Data

The Helioseismic and Magnetic Imager (HMI) and the Atmospheric Imaging Assembly (AIA) instruments onboard the Solar Dynamics Observatory satellite produce Doppler velocity and continuum intensity at 6173 A as well as intensity maps at 1600 A and 1700 A, which can be used for helioseismic studies at different heights in the solar photosphere. We perform a Hankel–Fourier analysis in an annulus centered around sunspots or quiet-Sun regions, to estimate the change in power of waves crossing these regions of interest. We find that there is a dependence of power-reduction coefficients α on measurement height in the photosphere: Sunspots reduce the power of outgoing waves with frequencies ν lower than ν≈4.5 mHz at all heights, but enhance the power of acoustic waves in the range ν≈4.5 – 5.5 mHz toward chromospheric heights, which is likely the signature of acoustic glories (halos). Maximum power reduction seems to occur near the continuum level and to decrease with altitude. Sunspots also impact the frequencies of outgoing waves in an altitude-dependent fashion. The quiet Sun is shown to behave like a strong power reducer for outgoing f- and p-modes at the continuum level, with a power reduction α≈15 – 20 %, and like a weak power enhancer for p-modes higher in the atmosphere. It is speculated that the surprising power reduction at the continuum level is related to granulation. In Doppler-velocity data, and unlike in intensity data, the quiet Sun behaves like a strong power reducer for granular flows.

Surface-Focused Seismic Holography of Sunspots: II. Expectations from Numerical Simulations Using Sound-Speed Perturbations

Helioseismic observations of sunspots show that wave travel times, at fixed horizontal phase speed, depend on the temporal frequency of the waves employed in the data analysis. This frequency variation has been suggested to be consistent with near-surface (vertical length scales of order one Mm or smaller) changes in wave propagation properties relative to the quiet Sun. We investigate this suggestion by employing numerical simulations of acoustic-wave propagation through models with horizontally and vertically inhomogeneous structure. Standard methods of surface-focused helioseismic holography are applied to the resulting simulated wave fields. We find that the travel-time shifts measured using holography from the simulations with deep sound-speed perturbations (relative to a plane-parallel quiet-Sun model) do not show a systematic frequency dependence at phase speeds above about 20 km s−1. However, shallow sound-speed perturbations, similar to those proposed to model the acoustic scattering properties of sunspots observed with Hankel analysis, produce systematic frequency dependence at these phase speeds. In both cases, positive travel-time shifts can be caused by positive sound-speed perturbations. The details of the travel-time shifts are, however, model dependent.

Genetic magnetohelioseismology with Hankel analysis data

Hankel analysis determined that sunspots absorb energy from and shift the phase of f- and p-modes incident upon them. One promising mechanism that can explain the absorption is partial conversion to slow magnetoacoustic-gravity waves and Alfv en waves, which guide energy along the magnetic eld away from the acoustic cavity. Our recent mode conversion calculations demonstrated that simple sunspot models, which roughly account for the radial variation of the magnetic eld strength and inclination, can produce ample absorption to explain the observations, along with phase shifts that agree remarkably well with the Hankel analysis data. In this paper, we follow the same approach, but adopt a more realistic model for the solar convection zone that includes the thermal perturbation associated with a sunspot’s magnetic eld. Consistent with our earlier ndings, we show that a moderately inclined, uniform magnetic eld exhibits signican tly enhanced absorption (mode conversion) in comparison to a vertical eld (depending on the frequency and radial order of the mode). A genetic algorithm is employed to adjust the parameters that control the radial structure of our sunspot models, in order to minimize the discrepancy between the theoretical predictions and the Hankel analysis measurements. For models that best t the phase shifts, the agreement with the Hankel analysis data is excellent, and the corresponding absorption coecien ts are generally in excess of the observed levels. On the other hand, for models that best t the phase shift and absorption data simultaneously the overall agreement is very good but the phase shifts agree less well. This is most likely caused by the dieren t size of the regions responsible for the absorption and phase shift. Typically, the eld strengths required by such models lie in the range 1 3 kG, compatible with observations for sunspots and active regions. While there remain some uncertainties, our results provide further evidence that mode conversion is the predominant mechanism responsible for the observed absorption in sunspots; and that eld inclination away from vertical is a necessary ingredient for any model that aims to simultaneously explain the phase shift and absorption data.

Observations and Interpretation of Subsurface Magnetic Structures

AbstractImmediate subsurface local solar structures – flow, magnetic, and thermal – especially those associated with active regions, are of considerable interest, and are potentially accessible using high ℓ helioseismology. In recent years, classical modal seismology has been supplemented with new approaches, such as time-distance helioseismology, Hankel analysis, and acoustic holography. Results from time-distance helioseismology on shallow subsurface wave speed are at odds with those from the other methods. A simple model is presented which yields good agreement with the observed Hankel phase shifts of p modes passing through sunspots.

Chromatic Holography of the Sunspot Acoustic Environment

We use helioseismic holography to obtain seismic images of a sunspot and its local environment over the 3-8 mHz spectrum. We are particularly interested in the deficit brought about by strong absorption by the sunspot itself in the 3-4 mHz range and in the helioseismic character of the acoustic recently discovered by Braun et al. The holographic images computed here clearly show that over a broad range in frequency the sunspot replaces the radiation impinging into it from the ambient solar interior with an outgoing flux that is only a fraction of that which it receives. This deficit persists uniformly over the 3-7 mHz spectrum, even as the reflectivity of the quiet-Sun photosphere goes from being an almost perfect, specular reflector at 4 mHz to an almost perfect absorber at 5 mHz. As far as we can judge, the moat surrounding the sunspot need not require a helioseismic absorption mechanism of its own. Its signature in 3-4 mHz images could arise from simple scattering of an deficit that originates in the nearby sunspot. Such scattering may be the result of a thermal perturbation resulting from the blockage of convective heat transport through the sunspot photosphere. Alternatively, it could be the signature of a Doppler perturbation attendant to the rapid convective outflow that might be driven by such a thermal accumulation. While the results presented here do not rule out the possibility that the moat has its own absorption mechanism, they show little independent evidence to indicate that the moat otherwise behaves very differently from the quiet Sun where absorption and reemission of flux are concerned. Helioseismic images of conspicuous halos that appear in 6 mHz power maps show no significant enhancement of emission from these regions. A fairly broad region surrounding the sunspot appears to render a weak enhancement, ~2.5%, in the local generation of 5 mHz power. This seems to explain peculiarities in the spectrum of flux balance measurements based on Hankel analysis. The distribution of the 5 mHz acoustic egression excess is fairly diffuse and does not seem to be spatially correlated with the strong power halos seen in 6 mHz power maps.