Convection in the solar interior is thought to comprise structures on a spectrum of scales. This conclusion emerges from phenomenological studies and numerical simulations, though neither covers the proper range of dynamical parameters of solar convection. Here, we analyze observations of the wavefield in the solar photosphere using techniques of time-distance helioseismology to image flows in the solar interior. We downsample and synthesize 900 billion wavefield observations to produce 3 billion cross-correlations, which we average and fit, measuring 5 million wave travel times. Using these travel times, we deduce the underlying flow systems and study their statistics to bound convective velocity magnitudes in the solar interior, as a function of depth and spherical-harmonic degree ℓ. Within the wavenumber band ℓ < 60, convective velocities are 20-100 times weaker than current theoretical estimates. This constraint suggests the prevalence of a different paradigm of turbulence from that predicted by existing models, prompting the question: what mechanism transports the heat flux of a solar luminosity outwards? Advection is dominated by Coriolis forces for wavenumbers ℓ < 60, with Rossby numbers smaller than approximately 10 −2 at r∕R ⊙ ¼ 0.96, suggesting that the Sun may be a much faster rotator than previously thought, and that large-scale convection may be quasi-geostrophic. The fact that isorotation contours in the Sun are not coaligned with the axis of rotation suggests the presence of a latitudinal entropy gradient.imaging | thermal wind balance | Reynolds stresses | inverse problem T he thin photosphere of the Sun, where thermal transport is dominated by free-streaming radiation, shows a spectrum in which granulation and supergranulation are most prominent. Observed properties of granules, such as spatial scales, radiative intensity, and photospheric spectral-line formation are successfully reproduced by numerical simulations (1, 2). In contrast, convection in the interior is not directly observable and likely governed by aspects more difficult to model, such as the integrity of descending plumes to diffusion and various instabilities (3). Further, solar convection is governed by extreme parameters (4) (Prandtl number approximately 10 −6 -10 −4 , Rayleigh number approximately 10 19 -10 24 , and Reynolds number approximately 10 12 -10 16 ), which make fully resolved three-dimensional direct numerical simulations impossible for the foreseeable future. It is likewise difficult to reproduce them in laboratory experiments.Turning to phenomenology, mixing-length theory (MLT) is predicated on the assumption that parcels of fluid of a specified spatial and velocity scale transport heat over one length scale (termed the mixing length) and are then mixed in the new environment. While this picture is simplistic (5), it has been successful in predicting aspects of solar structure as well as the dominant scale and magnitude of observed surface velocities. MLT posits a spatial convective scale that increases with depth ...

S U M M A R YWe determine finite-frequency sensitivity kernels for seismic interferometry based upon noise cross-correlation measurements. Under the assumptions that noise is spatially uncorrelated but non-uniform, we determine ensemble-averaged cross correlations between synthetic seismograms at two geographically distinct locations. By minimizing a measure of the difference between observed and simulated ensemble cross correlations-subject to the constraint that the simulated wavefield satisfies the seismic wave equation-we obtain ensemble sensitivity kernels. These ensemble kernels reflect the sensitivity of ensemble cross-correlation measurements to variations in model parameters, for example, mass density, shear and compressional wave speeds and the spatial distribution of noise. Ensemble kernels are calculated based upon the interaction between two wavefields: an ensemble forward wavefield and an ensemble adjoint wavefield. To obtain the ensemble forward wavefield, one first calculates a generating wavefield obtained by inserting a signal determined by the characteristics of the noise at the location of the first receiver, saving the results of this calculation at locations where noise is generated, that is, typically on (a portion of) the Earth's surface. Next, one uses this generating wavefield as the source of the ensemble forward wavefield associated with the first receiver. The ensemble adjoint wavefield is obtained by using measurements between simulated and observed ensemble cross correlations as a seismic source located at the second receiver. The interaction between ensemble forward and adjoint wavefields 'paints' ensemble sensitivity kernels. We illustrate the construction of ensemble kernels and their nature in two and three dimensions using a spectral-element method. In addition to a 'banana-doughnut' feature connecting the two receivers, as in traditional finite-frequency earthquake tomography, some noise cross-correlation sensitivity kernels exhibit hyperbolic 'jets' protruding from each receiver in a direction away from the other receiver. Ensemble sensitivity kernels for long-period (T > ∼50 s) non-uniform noise in global models exhibit sensitivity along the minor and major arcs. These kernels reflect the fact that measurements typically involve long time-series that include multi-orbit surface waves. Like free oscillations, such measurements are sensitive to structure along the great circle through the two receivers. From the perspective of noise cross-correlation tomography, we discuss strategies for inversions in terrestrial and helioseismology.

Various methods of helioseismology are used to study the subsurface properties of the sunspot in NOAA Active Region 9787. This sunspot was chosen because it is axisymmetric, shows little evolution during 20-28 January 2002, and was observed continuously by the MDI/SOHO instrument. AR 9787 is visible on helioseismic maps of the farside of the Sun from 15 January, i.e. days before it crossed the East limb.Oscillations have reduced amplitudes in the sunspot at all frequencies, whereas a region of enhanced acoustic power above 5.5 mHz (above the quiet-Sun acoustic cutoff) is seen outside the sunspot and the plage region. This enhanced acoustic power has been suggested to be caused by the conversion of acoustic waves into magneto-acoustic waves that are refracted back into the interior and re-emerge as acoustic waves in the quiet Sun. Observations show that the sunspot absorbs a significant fraction of the incoming p and f modes around 3 mHz. A numerical simulation of MHD wave propagation through a simple model of AR 9787 confirmed that wave absorption is likely to be due to the partial conversion of incoming waves into magneto-acoustic waves that propagate down the sunspot. Wave travel times and mode frequencies are affected by the sunspot. In most cases, wave packets that propagate through the sunspot have reduced travel times. At short travel distances, however, the sign of the travel-time shifts appears to depend sensitively on how the data are processed and, in particular, on filtering in frequency-wavenumber space. We carry out two linear inversions for wave speed: one using travel-times and phase-speed filters and the other one using mode frequencies from ring analysis. These two inversions give subsurface wave-speed profiles with opposite signs and different amplitudes.The travel-time measurements also imply different subsurface flow patterns in the surface layer depending on the filtering procedure that is used. Current sensitivity kernels are unable to reconcile these measurements, perhaps because they rely on imperfect models of the power spectrum of solar oscillations. We present a linear inversion for flows of ridge-filtered travel times. This inversion shows a horizontal outflow in the upper 4 Mm that is consistent with the moat flow deduced from the surface motion of moving magnetic features.From this study of AR 9787, we conclude that we are currently unable to provide a unified description of the subsurface structure and dynamics of the sunspot.

The differentially rotating outer layers of stars are thought to play a role in driving their magnetic activity, but the underlying mechanisms that generate and sustain differential rotation are poorly understood. We report the measurement using asteroseismology of latitudinal differential rotation in the convection zones of 40 Sun-like stars. For the most significant detections, the stars’ equators rotate approximately twice as fast as their midlatitudes. The latitudinal shear inferred from asteroseismology is much larger than predictions from numerical simulations.