Solar Rossby waves observed in GONG++ ring-diagram flow maps

Context. Solar Rossby waves have only recently been unambiguously identified in Helioseimsic and Magnetic Imager (HMI) and Michelson Doppler Imager maps of flows near the solar surface. So far this has not been done with the Global Oscillation Network Group (GONG) ground-based observations, which have different noise properties. Aims. We use 17 years of GONG++ data to identify and characterize solar Rossby waves using ring-diagram helioseismology. We compare directly with HMI ring-diagram analysis. Methods. Maps of the radial vorticity were obtained for flows within the top 2 Mm of the surface for 17 years of GONG++ data. The data were corrected for systematic effects including the annual periodicity related to the B0 angle. We then computed the Fourier components of the radial vorticity of the flows in the co-rotating frame. We performed the same analysis on the HMI data that overlap in time. Results. We find that the solar Rossby waves have measurable amplitudes in the GONG++ sectoral power spectra for azimuthal orders between m = 3 and m = 15. The measured mode characteristics (frequencies, lifetimes, and amplitudes) from GONG++ are consistent with the HMI measurements in the overlap period from 2010 to 2018 for m ≤ 9. For higher-m modes the amplitudes and frequencies agree within two sigmas. The signal-to-noise ratio of modes in GONG++ power spectra is comparable to those of HMI for 8 ≤ m ≤ 11, but is lower by a factor of two for other modes. Conclusions. The GONG++ data provide a long and uniform data set that can be used to study solar global-scale Rossby waves from 2001.

Exploring the latitude and depth dependence of solar Rossby waves using ring-diagram analysis

Context. Global-scale equatorial Rossby waves have recently been unambiguously identified on the Sun. Like solar acoustic modes, Rossby waves are probes of the solar interior. Aims. We study the latitude and depth dependence of the Rossby wave eigenfunctions. Methods. By applying helioseismic ring-diagram analysis and granulation tracking to observations by HMI aboard SDO, we computed maps of the radial vorticity of flows in the upper solar convection zone (down to depths of more than 16 Mm). The horizontal sampling of the ring-diagram maps is approximately 90 Mm (∼7.5°) and the temporal sampling is roughly 27 hr. We used a Fourier transform in longitude to separate the different azimuthal orders m in the range 3 ≤ m ≤ 15. At each m we obtained the phase and amplitude of the Rossby waves as functions of depth using the helioseismic data. At each m we also measured the latitude dependence of the eigenfunctions by calculating the covariance between the equator and other latitudes. Results. We conducted a study of the horizontal and radial dependences of the radial vorticity eigenfunctions. The horizontal eigenfunctions are complex. As observed previously, the real part peaks at the equator and switches sign near ±30°, thus the eigenfunctions show significant non-sectoral contributions. The imaginary part is smaller than the real part. The phase of the radial eigenfunctions varies by only ±5° over the top 15 Mm. The amplitude of the radial eigenfunctions decreases by about 10% from the surface down to 8 Mm (the region in which ring-diagram analysis is most reliable, as seen by comparing with the rotation rate measured by global-mode seismology). Conclusions. The radial dependence of the radial vorticity eigenfunctions deduced from ring-diagram analysis is consistent with a power law down to 8 Mm and is unreliable at larger depths. However, the observations provide only weak constraints on the power-law exponents. For the real part, the latitude dependence of the eigenfunctions is consistent with previous work (using granulation tracking). The imaginary part is smaller than the real part but significantly nonzero.

An improved multi-ridge fitting method for ring-diagram helioseismic analysis

Context. There is a wide discrepancy in current estimates of the strength of convection flows in the solar interior obtained using different helioseismic methods applied to observations from the Helioseismic and Magnetic Imager onboard the Solar Dynamics Observatory. The cause for these disparities is not known. Aims. As one step in the effort to resolve this discrepancy, we aim to characterize the multi-ridge fitting code for ring-diagram helioseismic analysis that is used to obtain flow estimates from local power spectra of solar oscillations. Methods. We updated the multi-ridge fitting code developed by Greer et al. (2014, Sol. Phys., 289, 2823) to solve several problems we identified through our inspection of the code. In particular, we changed the (1) merit function to account for the smoothing of the power spectra, (2) model for the power spectrum, and (3) noise estimates. We used Monte Carlo simulations to generate synthetic data and to characterize the noise and bias of the updated code by fitting these synthetic data. Results. The bias in the output fit parameters, apart from the parameter describing the amplitude of the p-mode resonances in the power spectrum, is below what can be measured from the Monte-Carlo simulations. The amplitude parameters are underestimated; this is a consequence of choosing to fit the logarithm of the averaged power. We defer fixing this problem as it is well understood and not significant for measuring flows in the solar interior. The scatter in the fit parameters from the Monte-Carlo simulations is well-modeled by the formal error estimates from the code. Conclusions. We document and demonstrate a reliable multi-ridge fitting method for ring-diagram analysis. The differences between the updated fitting results and the original results are less than one order of magnitude and therefore we suspect that the changes will not eliminate the aforementioned orders-of-magnitude discrepancy in the amplitude of convective flows in the solar interior.

Time-distance helioseismology of solar Rossby waves

Context. Solar Rossby waves (r modes) have recently been discovered in the near-surface horizontal flow field using the techniques of granulation-tracking and ring-diagram analysis applied to six years of SDO/HMI data. Aims. Here we apply time-distance helioseismology to the combined SOHO/MDI and SDO/HMI data sets, which cover 21 years of observations from May 1996 to April 2017. The goal of this study is to provide an independent confirmation over two solar cycles and in deeper layers of the Sun. Methods. We have measured south-north helioseismic travel times along the equator, which are sensitive to subsurface north-south flows. To reduce noise, the travel times were averaged over travel distances from 6° to 30°; the mean distance corresponds to a p-mode lower turning point of 0.91 R⊙. The 21-year time series of travel-time measurements was split into three seven-year subsets and transformed to obtain power spectra in a corotating frame. Results. The power spectra all show peaks near the frequencies of the classical sectoral Rossby waves for azimuthal wavenumbers in the range 3 ≤ m ≤ 15. The mode frequencies and linewidths of the modes with m ≤ 9 are consistent with a previous study whereas modes with m ≥ 10 are shifted toward less negative frequencies by 10–20 nHz. While most of these modes have e-folding lifetimes on the order of a few months, the longest lived mode, m = 3, has an e-folding lifetime of more than one year. For each mode, the rms velocity at the equator is in the range of 1–3 m s−1, with the largest values for m ∼ 10. No evidence for the m = 2 sectoral mode is found in the power spectrum, implying that the rms velocity of this mode is below ∼0.5 m s−1. Conclusions. This work confirms the existence of equatorial global Rossby waves in the solar interior over the past two solar cycles and shows that time-distance helioseismology is a promising technique to study them deep in the convection zone.

Supervised neural networks for helioseismic ring-diagram inversions

Context. The inversion of ring fit parameters to obtain subsurface flow maps in ring-diagram analysis for eight years of SDO observations is computationally expensive, requiring ∼3200 CPU hours. Aims. In this paper we apply machine-learning techniques to the inversion step of the ring diagram pipeline in order to speed up the calculations. Specifically, we train a predictor for subsurface flows using the mode fit parameters and the previous inversion results to replace future inversion requirements. Methods. We utilize artificial neural networks (ANNs) as a supervised learning method for predicting the flows in 15° ring tiles. We discuss each step of the proposed method to determine the optimal approach. In order to demonstrate that the machine-learning results still contain the subtle signatures key to local helioseismic studies, we use the machine-learning results to study the recently discovered solar equatorial Rossby waves. Results. The ANN is computationally efficient, able to make future flow predictions of an entire Carrington rotation in a matter of seconds, which is much faster than the current ∼31 CPU hours. Initial training of the networks requires ∼3 CPU hours. The trained ANN can achieve a rms error equal to approximately half that reported for the velocity inversions, demonstrating the accuracy of the machine learning (and perhaps the overestimation of the original errors from the ring-diagram pipeline). We find the signature of equatorial Rossby waves in the machine-learning flows covering six years of data, demonstrating that small-amplitude signals are maintained. The recovery of Rossby waves in the machine-learning flow maps can be achieved with only one Carrington rotation (27.275 days) of training data. Conclusions. We show that machine learning can be applied to and perform more efficiently than the current ring-diagram inversion. The computation burden of the machine learning includes 3 CPU hours for initial training, then around 10−4 CPU hours for future predictions.

Asymmetry in Solar Torsional Oscillation

Solar torsional oscillations are migrating bands of slower and faster than average rotation, which are thought to be related to the Sun’s magnetic cycle. We perform the first long-term study (16 years) of hemispherical asymmetry in solar torsional oscillation velocity using helioseismic data. We explore the spatial and temporal variation of North-South asymmetry using zonal flow velocities obtained from ring diagram analysis of the Global Oscillation Network Group (GONG) Doppler images. We find a strong correlation between the asymmetries of near-surface torsional oscillation with magnetic flux and sunspot number, with the velocity asymmetry preceding in both the cases. We speculate that the asymmetry in torsional oscillation velocity may help in predicting the hemispherical asymmetry in the sunspot cycle.