Anomalously weak solar convection

Abstract

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([symbol: see text]) = 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.

Fig. 1.

Line-of-sight Doppler velocities are measured every 45 s at 4,096 × 4,096 pixels on the solar photosphere by the Helioseismic and Magnetic Imager (background image). We cross correlate wavefield records of temporal length T at points on opposing quadrants (blue with blue or red with red). These “blue” and “red” correlations are separately averaged, respectively sensitive to longitudinal and latitudinal flow at (θ,ϕ; r/R_⊙ = 0.96), where (θ,ϕ) is the central point marked by a cross (see Fig. 2 for further illustration). The longitudinal measurement is sensitive to flows in that direction while the latitudinal measurement to flows along latitude. We create a travel-time maps δτ(θ,ϕ,T) by making this measurement about various central points (θ,ϕ) on the surface. Each travel time is obtained upon correlating the wavefield between 600 pairs of points distributed in azimuth.

Fig. 2.

The cross-correlation measurement geometry (upper box; arrowheads—horizontal: longitude, and vertical: latitude) used to image the layer r/R_⊙ = 0.96 (dot-dashed line). Doppler velocities of temporal length T measured at the solar surface are cross correlated between point pairs at opposite ends of annular discs (colored red and blue); e.g., points on the innermost blue sector on the left are correlated with diagonally opposite points on the outermost blue sector on the right. Six-hundred correlations are prepared and averaged for each travel-time measurement. Travel times of waves that propagate along paths in the direction of the horizontal and vertical arrows are primarily sensitive to longitudinal and latitudinal flows, v_ϕ and v_θ, respectively. The focus point of these waves is at r/R_⊙ = 0.96 (lower box) and the measured travel-time shift δτ(θ,ϕ,T) is linearly related to the flow component v(r/R_⊙ = 0.96,θ,ϕ) with a contribution from the incoherent wave noise. We are thus able to map the flow field at specific depths v(r,θ,ϕ) through appropriate measurements of δτ(θ,ϕ,T). For the inversions here, we create travel-time maps of size 128 × 128 (see Fig. 3). For reference, we note that the base of the convection zone is located at r/R_⊙ = 0.71 and the near-surface shear layer extends from r/R_⊙ = 0.9 upwards.

Fig. 3.

A travel-time map consisting 16,384 travel-time measurements, spanning a 60° × 60° region (at a resolution of 0.46875 ° per pixel) around the solar disk center, obtained by analyzing one day’s worth of data taken by the Helioseismic and Magnetic Imager instrument (14) onboard the Solar Dynamics Observatory satellite. 3.2 billion wavefield measurements were analyzed to generate 10 million correlations, which were averaged and fitted to generate this travel-time map. This geometry and these particular wave times are so chosen as to be sensitive to flow systems in the solar interior. The spectrum of these travel times shows no interesting or anomalous peaks that meet the detection criteria (described subsequently).

Fig. 4.

Because wavelengths of helioseismic waves may be comparable to or larger than convective features through which they propagate, the ray approximation is inaccurate and finite-wavelength effects must be accounted for when modeling wave propagation in the Sun (20). In order to derive the 3D finite-frequency sensitivity function (kernel) associated with a travel-time measurement (21), we simulate waves propagating through a randomly scattered set of 500 east-west-flow ‘delta’ functions, each of which is assigned a random sign so as not to induce a net flow signal (22) (upper box). We place these flow deltas in a latitudinal band of extent 120° centered about the equator, because the quality of observational data degrades outside of this region. We perform six simulations, with these deltas placed at a different depth in each instance, so as to sample the kernel at these radii. The bottom four boxes show slices at various radii of the sensitivity function for the measurement which attempts to resolve flows at r/R_⊙ = 0.96. Measurement sensitivity is seen to peak at the focus depth, a desirable quality, but contains near-surface lobes as well. Note that the volume integral of flows in the solar interior with this kernel function gives rise to the associated travel-time shift, which explains the units.

Fig. 5.

Observational bounds on flow magnitudes and the associated Rossby numbers. Boxes (A, B): solid curves with 1-σ error bars (standard deviations) show observational constraints on lateral flows averaged over m at radial depths, r/R_⊙ = 0.92, 0.96; dot-dash lines are spectra from ASH convection simulations (6). Colors differentiate between the focus depth of the measurement and coherence times. At a depth of r/R_⊙ = 0.96, simulations of convection (6) show a coherence time of T_coh = 24 hours (A) while MLT (16) gives T_coh = 96 hours (B), the latter obtained by dividing the mixing length by the predicted velocity. Both MLT and simulations (23, 24) indicate a convective depth coherence over 1.8 pressure scale heights, an input to our inversion. At r/R_⊙ = 0.96, MLT predicts a 60 ms^-1, ℓ = 61 convective flow and for r/R_⊙ = 0.92, an ℓ = 33, 45 ms^-1 flow [upon applying continuity considerations (23)]. (C) shows upper bounds on Rossby number, Ro = U/(2ΩL), , r = 0.92, 0.96 R_⊙. Interior convection appears to be strongly geostrophically balanced (i.e., rotationally dominated) on these scales. By construction, these measurements are sensitive to lateral flows i.e., longitudinal and latitudinal at these specific depths (r/R_⊙ = 0.92, 0.96) and consequently, we denote these flow components (longitudinal or latitudinal) by scalars.