Helioseismology challenges models of solar convection

Convection is the mechanism by which energy is transported through the outermost 30% of the Sun. Solar turbulent convection is notoriously difficult to model across the entire convection zone where the density spans many orders of magnitude. In this issue of PNAS, Hanasoge et al. (2012) employ recent helioseismic observations to derive stringent empirical constraints on the amplitude of large-scale convective velocities in the solar interior. They report an upper limit that is far smaller than predicted by a popular hydrodynamic numerical simulation.Comment: Printed in the Proceedings of the National Academy of Sciences (2 pages, 1 figure). Available at http://www.pnas.org/cgi/doi/10.1073/pnas.120887510

Propagation of seismic waves through a spatio-temporally fluctuating medium: Homogenization

Measurements of seismic wave travel times at the photosphere of the Sun have enabled inferences of its interior structure and dynamics. In interpreting these measurements, the simplifying assumption that waves propagate through a temporally stationary medium is almost universally invoked. However, the Sun is in a constant state of evolution, on a broad range of spatio-temporal scales. At the zero wavelength limit, i.e., when the wavelength is much shorter than the scale over which the medium varies, the WKBJ (ray) approximation may be applied. Here, we address the other asymptotic end of the spectrum, the infinite wavelength limit, using the technique of homogenization. We apply homogenization to scenarios where waves are propagating through rapidly varying media (spatially and temporally), and derive effective models for the media. One consequence is that a scalar sound speed becomes a tensorial wavespeed in the effective model and anisotropies can be induced depending on the nature of the perturbation. The second term in this asymptotic two-scale expansion, the so-called corrector, contains contributions due to higher-order scattering, leading to the decoherence of the wavefield. This decoherence may be causally linked to the observed wave attenuation in the Sun. Although the examples we consider here consist of periodic arrays of perturbations to the background, homogenization may be extended to ergodic and stationary random media. This method may have broad implications for the manner in which we interpret seismic measurements in the Sun and for modeling the effects of granulation on the scattering of waves and distortion of normal-mode eigenfunctions.Comment: 17 pages, 6 figures, in press, Ap

Problems in computational helioseismology

We discuss current advances in forward and inverse modeling for local helioseismology. We report theoretical uniqueness results, in particular the Novikov-Agaltsov reconstruction algorithm, which is relevant to solving the non-linear inverse problem of time-distance helioseismology (finite amplitude pertubations to the medium). Numerical experiments were conducted to determine the number of frequencies required to reconstruct density and sound speed in the solar interior.Comment: Oberwolfach Report, Computational Inverse Problems for Partial Differential Equations, 14 May - 20 May 2017. https://www.mfo.de/occasion/1720/www_vie

Solar-cycle variation of the rotational shear near the solar surface

Helioseismology has revealed that the angular velocity of the Sun increases with depth in the outermost 35 Mm of the Sun. Recently, we have shown that the logarithmic radial gradient ($\rm d\ln\Omega/\rm d\ln r$) in the upper 10~Mm is close to $-1$ from the equator to $60^\circ$ latitude.We aim to measure the temporal variation of the rotational shear over solar cycle 23 and the rising phase of cycle 24 (1996-2015). We used f mode frequency splitting data spanning 1996 to 2011 from the Michelson Doppler Imager (MDI) and 2010 to 2015 from the Helioseismic Magnetic Imager (HMI). In a first for such studies, the f mode frequency splitting data were obtained from 360-day time series. We used the same method as in our previous work for measuring $\rm d\ln\Omega/d\ln r$ from the equator to $80^\circ$ latitude in the outer 13~Mm of the Sun. Then, we calculated the variation of the gradient at annual cadence relative to the average over 1996 to 2015. We found the rotational shear at low latitudes ($0^\circ$ to $30^\circ$) to vary in-phase with the solar activity, varying by $\sim \pm 10$\% over the period 1996 to 2015. At high latitudes ($60^\circ$ to $80^\circ$), we found rotational shear to vary in anti-phase with the solar activity. By comparing the radial gradient obtained from the splittings of the 360-day and the corresponding 72-day time series of HMI and MDI data, we suggest that the splittings obtained from the 72-day HMI time series suffer from systematic errors. We provide a quantitative measurement of the temporal variation of the outer part of the near surface shear layer which may provide useful constraints on dynamo models and differential rotation theory.Comment: 5 pages, 6 figure

Interpretation of Helioseismic Travel Times - Sensitivity to Sound Speed, Pressure, Density, and Flows

Time-distance helioseismology uses cross-covariances of wave motions on the solar surface to determine the travel times of wave packets moving from one surface location to another. We review the methodology to interpret travel-time measurements in terms of small, localized perturbations to a horizontally homogeneous reference solar model. Using the first Born approximation, we derive and compute 3D travel-time sensitivity (Fr\'echet) kernels for perturbations in sound-speed, density, pressure, and vector flows. While kernels for sound speed and flows had been computed previously, here we extend the calculation to kernels for density and pressure, hence providing a complete description of the effects of solar dynamics and structure on travel times. We treat three thermodynamic quantities as independent and do not assume hydrostatic equilibrium. We present a convenient approach to computing damped Green's functions using a normal-mode summation. The Green's function must be computed on a wavenumber grid that has sufficient resolution to resolve the longest lived modes. The typical kernel calculations used in this paper are computer intensive and require on the order of 600 CPU hours per kernel. Kernels are validated by computing the travel-time perturbation that results from horizontally-invariant perturbations using two independent approaches. At fixed sound-speed, the density and pressure kernels are approximately related through a negative multiplicative factor, therefore implying that perturbations in density and pressure are difficult to disentangle. Mean travel-times are not only sensitive to sound-speed, density and pressure perturbations, but also to flows, especially vertical flows. Accurate sensitivity kernels are needed to interpret complex flow patterns such as convection

Fragile detection of solar g modes by Fossat et al

The internal gravity modes of the Sun are notoriously difficult to detect, and the claimed detection of gravity modes presented in Fossat et al. 2017 is thus very exciting. Given the importance of these modes for understanding solar structure and dynamics, the results must be robust. While Fossat et al. 2017 described their method and parameter choices in detail, the sensitivity of their results to several parameters were not presented. Therefore, we test the sensitivity to a selection of them. The most concerning result is that the detection vanishes when we adjust the start time of the 16.5 year velocity time series by a few hours. We conclude that this reported detection of gravity modes is extremely fragile and should be treated with utmost caution.Comment: 15 pages, 11 Figure

Revisiting the exomoon candidate signal around Kepler-1625b

Transit photometry of the exoplanet candidate Kepler-1625b has recently been interpreted to show hints of a moon. We aim to clarify whether the exomoon-like signal is really caused by a large object in orbit around Kepler-1625b. We explore several detrending procedures, i.e. polynomials and the Cosine Filtering with Autocorrelation Minimization (CoFiAM). We then supply a light curve simulator with the co-planar orbital dynamics of the system and fit the resulting planet-moon transit light curves to the Kepler data. We employ the Bayesian Information Criterion (BIC) to assess whether a single planet or a planet-moon system is a more likely interpretation of the light curve variations. We carry out a blind hare-and-hounds exercise using many noise realizations by injecting simulated transits into different out-of-transit parts of the original Kepler-1625 data: 100 sequences with 3 synthetic transits of a Kepler-1625b-like planet and 100 sequences with 3 synthetic transits of this planet with a Neptune-sized moon. The statistical significance and characteristics of the exomoon-like signal strongly depend on the detrending method, and the data chosen for detrending, and on the treatment of gaps in the light curve. Our injection-retrieval experiment shows evidence for moons in about 10% of those light curves that do not contain an injected moon. Strikingly, many of these false-positive moons resemble the exomoon candidate. We recover up to about half of the injected moons, depending on the detrending method, with radii and orbital distances broadly corresponding to the injected values. A $\Delta$BIC of -4.9 for the CoFiAM-based detrending indicates an exomoon around Kepler-1625b. This solution, however, is only one out of many and we find very different solutions depending on the details of the detrending method. It is worrying that the detrending is key to the interpretation of the data.Comment: 16 pages, 12 figures. Accepted for publication by A&

Generalization of the noise model for time-distance helioseismology

In time-distance helioseismology, information about the solar interior is encoded in measurements of travel times between pairs of points on the solar surface. Travel times are deduced from the cross-covariance of the random wave field. Here we consider travel times and also products of travel times as observables. They contain information about e.g. the statistical properties of convection in the Sun. The basic assumption of the model is that noise is the result of the stochastic excitation of solar waves, a random process which is stationary and Gaussian. We generalize the existing noise model (Gizon and Birch 2004) by dropping the assumption of horizontal spatial homogeneity. Using a recurrence relation, we calculate the noise covariance matrices for the moments of order 4, 6, and 8 of the observed wave field, for the moments of order 2, 3 and 4 of the cross-covariance, and for the moments of order 2, 3 and 4 of the travel times. All noise covariance matrices depend only on the expectation value of the cross-covariance of the observed wave field. For products of travel times, the noise covariance matrix consists of three terms proportional to $1/T$, $1/T^2$, and $1/T^3$, where $T$ is the duration of the observations. For typical observation times of a few hours, the term proportional to $1/T^2$ dominates and $Cov[\tau_1 \tau_2, \tau_3 \tau_4] \approx Cov[\tau_1, \tau_3] Cov[\tau_2, \tau_4] + Cov[\tau_1, \tau_4] Cov[\tau_2, \tau_3]$, where the $\tau_i$ are arbitrary travel times. This result is confirmed for $p_1$ travel times by Monte Carlo simulations and comparisons with SDO/HMI observations. General and accurate formulae have been derived to model the noise covariance matrix of helioseismic travel times and products of travel times. These results could easily be generalized to other methods of local helioseismology, such as helioseismic holography and ring diagram analysis