The Sun’s magnetic field is generated by subsurface motions of the convecting plasma. The latitude at which the magnetic field emerges through the solar surface (as sunspots) drifts toward the equator over the course of the 11-year solar cycle. We use helioseismology to infer the meridional flow (in the latitudinal and radial directions) over two solar cycles covering 1996–2019. Two data sources are used, which agree during their overlap period of 2001–2011. The time-averaged meridional flow is shown to be a single cell in each hemisphere, carrying plasma toward the equator at the base of the convection zone with a speed of ~4 meters per second at 45° latitude. Our results support the flux-transport dynamo model, which explains the drift of sunspot-emergence latitudes through the meridional flow.

Context. Local helioseismology has so far relied on semi-analytical methods to compute the spatial sensitivity of wave travel times to perturbations in the solar interior. These methods are cumbersome and lack flexibility. Aims. Here we propose a convenient framework for numerically solving the forward problem of time-distance helioseismology in the frequency domain. The fundamental quantity to be computed is the cross-covariance of the seismic wavefield. Methods. We choose sources of wave excitation that enable us to relate the cross-covariance of the oscillations to the Green's function in a straightforward manner. We illustrate the method by considering the 3D acoustic wave equation in an axisymmetric reference solar model, ignoring the effects of gravity on the waves. The symmetry of the background model around the rotation axis implies that the Green's function can be written as a sum of longitudinal Fourier modes, leading to a set of independent 2D problems. We use a high-order finite-element method to solve the 2D wave equation in frequency space. The computation is 'embarrassingly parallel', with each frequency and each azimuthal order solved independently on a computer cluster. Results. We compute travel-time sensitivity kernels in spherical geometry for flows, sound speed, and density perturbations under the first Born approximation. Convergence tests show that travel times can be computed with a numerical precision better than one millisecond, as required by the most precise travel-time measurements. Conclusions. The method presented here is computationally efficient and will be used to interpret travel-time measurements in order to infer, e.g., the large-scale meridional flow in the solar convection zone. It allows the implementation of (full-waveform) iterative inversions, whereby the axisymmetric background model is updated at each iteration.

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Context. Solar sectoral Rossby waves have only recently been unambiguously identified in Helioseimsic and Magnetic Imager (HMI) and Michelson Doppler Imager (MDI) 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 utilize 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 are obtained for flows within the top 2 Mm of the surface for 17 years of GONG++. The data is corrected for systematic effects including the annual periodicity related to the B 0 angle. We then compute the Fourier components of the radial vorticity of the flows in the co-rotating frame. We perform 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 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 to study solar global-scale Rossby waves from 2001.

The temporal covariance between seismic waves measured at two locations on the solar surface is the fundamental observable in time-distance helioseismology. Above the acoustic cut-off frequency (∼5.3 mHz), waves are not trapped in the solar interior and the covariance function can be used to probe the upper atmosphere. We wish to implement appropriate radiative boundary conditions for computing the propagation of high-frequency waves in the solar atmosphere. We consider the radiative boundary conditions recently developed by Barucq et al. (2017) for atmospheres in which sound-speed is constant and density decreases exponentially with radius. We compute the cross-covariance function using a finite element method in spherical geometry and in the frequency domain. The ratio between first-and second-skip amplitudes in the time-distance diagram is used as a diagnostic to compare boundary conditions and to compare with observations. We find that a boundary condition applied 500 km above the photosphere and derived under the approximation of small angles of incidence accurately reproduces the 'infinite atmosphere' solution for high-frequency waves. When the radiative boundary condition is applied 2 Mm above the photosphere, we find that the choice of atmospheric model affects the time-distance diagram. In particular, the time-distance diagram exhibits double-ridge structure when using a VAL atmospheric model.