Turbulence in the Sun is suppressed on large scales and confined to equatorial regions

Abstract

Convection in the Sun's outer envelope generates turbulence and drives differential rotation, meridional circulation, and the global magnetic cycle. We develop a greater understanding of these processes by contrasting observations with simulations of global convection. These comparisons also enhance our comprehension of the physics of distant Sun-like stars. Here, we infer toroidal flow power as a function of wave number, frequency, and depth in the solar interior through helioseismic analyses of space-based observations. The inferred flows grow with spatial wave number and temporal frequency and are confined to low latitudes, supporting the argument that rotation induces systematic differences between the poles and equator. In contrast, the simulations used here show the opposite trends-power diminishing with increasing wave number and frequency while flow amplitudes become weakest at low latitudes. These differences highlight gaps in our understanding of solar convection and point to challenges ahead.

Fig. 1. Rotational shear obtained from simulations.

A snapshot of computed convection (A) and simulated and observed differential rotation (B and C) (38). Although Taylor-Proudman columns appear at low latitudes in the simulations, the amplitudes and contours compare favorably. (D) Numerical simulations of convection (4) show that toroidal $(\sqrt{{\sum }_{s,t,\mathrm{\sigma }}s(s+1){\mid w_{\mathit{st}}^{\mathrm{\sigma }}\mid }^2})$ and poloidal flow root mean square (RMS) velocities $(\sqrt{{\sum }_{s,t,\mathrm{\sigma }}{\mid u_{\mathrm{st}}^{\mathrm{\sigma }}\mid }^2+s(s+1){\mid v_{\mathrm{st}}^{\mathrm{\sigma }}\mid }^2})$, where u, v, and w are defined in Eq. 1, are comparable at all radii. Poloidal flows are compressible and have both radial and lateral components, whereas toroidal flows, by construction, are solely lateral and are incompressible. As a consequence, poloidal flows are the carriers of thermal flux; the precise role of toroidal flows in this picture is, however, not well understood. The similarity in magnitudes between the flows suggests that they likely play a critical role in regulating solar convection.

Fig. 2. B-coefficient power (blue solid lines) at r/R_⊙ = 0.995, measured from 8 years of HMI data (each year provides one inference and this plot therefore contains eight curves) and systematic background (red dashed line) computed from the model in eqs. S15 and S16.

The frequencies and other mode parameter fits vary from one 360-day period to another, which is why the noise background power shows dispersion. The theoretically computed noise background follows the observed B-coefficient power closely except at low s − ∣t∣, where the observed B-coefficient power is significantly larger than the model. This additionally gives us confidence that our models for the noise are reasonable.

Fig. 3. Toroidal-flow velocity amplitude, ∑t,σs(s+1)|wstσ(r0)|2 summed over the range 0 < σ ≤ 3.05 μHz (see Fourier convention in eq. S48 of section S6), is plotted as a function of harmonic degree s.

Shaded areas in (A to C) mark 1-σ errors in velocity. The legend in (A) applies to (B) as well. In contrast to the simulation, where velocity power peaks at s ≈ 15 and continuously decreases in either end, we see in (A to C) that power weakens with decreasing wave number (s), indicating that large-scale toroidal flows are suppressed. The comparison in (C) between toroidal flow amplitudes measured directly from surface Doppler measurements (33) and seismic inferences at r/R_⊙ = 0.995 is favorable; the simulated power does not make an appearance since the upper boundary of the computational domain is r/R_⊙ = 0.94. (D to E) Inferred RMS velocity compares reasonably with simulations [in contrast to prior estimates (5)]. Horizontal bars denote the width of the averaging kernel in (D to E). (F) RMS ratios of simulated-to-observed velocity as functions of harmonic degree s at different depths. Vertical error bars are computed using the variance in the spectra obtained from each of the eight 360-day periods analyzed here.

Fig. 4. Toroidal flow spectra (shaded lines) plotted as a function of s − ∣t∣.

The legend in (A) applies to (B) as well. Low s − ∣t∣ correspond to sectoral harmonics, i.e., equatorially localized north-south–aligned features, whereas harmonics with large s − ∣t∣ are present at all latitudes. Having obtained flow coefficients $w_{\mathrm{st}}^{\mathrm{\sigma }}$, we reconstruct the flow in the spatiotemporal domain using Eq. 1, for radii r/R_⊙ = 0.87 (A) and r/R_⊙ = 0.94 (B). (C) Longitudinally and temporally averaged simulated flow velocity as a function of latitude. (D) Corresponding latitudinal distribution power of the Sun at radius r/R_⊙ = 0.94. Seismically inferred flows are well confined near the equator, whereas simulations peak at higher latitudes. The thin vertical lines in (D) mark full widths at half maxima of v_θ and v_ϕ.

Fig. 5

Instantaneous snapshot of toroidal flow, reconstructed from the inferred (A and B) and simulated (C and D) $w_{\mathrm{st}}^{\mathrm{\sigma }}$ using the third term of Eq. 1, shown in the Mollweide projection. Since we do not have access to the even-s channels here and do not know the phases of the background systematics (we only model the expected power spectrum of systematics), we are unable to recover the true toroidal flow field from observations. However, since the signal at low s − ∣t∣ is substantially larger than the background noise model, some of the low-latitude features in (A) and (B) that are seen here are possibly real. Equatorially confined, north-south–aligned features are seen in both longitudinal [B; v_ϕ = − ∂_θw^σ(r, θ, ϕ)] and latitudinal velocity [A; v_θ = (sin θ)⁻¹ ∂_ϕw^σ(r, θ, ϕ)]. Small scales are preferentially enhanced in the inferred velocity image because the power spectrum steadily increases with wave number (Fig. 3, A and B). In contrast, the geometry of simulated flows (C and D) shows features of larger scales, very different from that of the inferences, because the power spectral shapes are different (Fig. 3). Both inferences and simulations are normalized to facilitate comparison.

Fig. 6. Toroidal flow spectra plotted as a function of frequency.

The spectrum $\sqrt{{\sum }_{s,t}s(s+1){\mid w_{\mathit{st}}^{\mathrm{\sigma }}(r_0)\mid }^2}$ steadily increases with temporal frequency σ (A), in contrast with simulations (C). In this low-frequency range, the velocity power in the Sun continues to grow much more rapidly with frequency σ than in numerical simulations, eventually becoming larger. Some of this could be arising from the low–spatial frequency tail of supergranulation, although temporally, we are inferring power in the range where supergranules have very little power (5). (B) Spatiotemporal power spectrum of sectoral modes (s = ∣t∣) of convection at r/R_⊙ = 0.95 as a function of frequency σ and azimuthal wave number t. These sectoral modes, which tend to be focused in the equatorial regions, correspond to the low-latitude features of Figs. 4 and 5. Material flows are suppressed by magnetic fields (34), thereby suggesting that toroidal power could be anticorrelated with the solar cycle. However, (D) suggests that surface and interior velocities (dotted and solid lines, respectively) are uncorrelated with the solar magnetic cycle (the sunspot number is depicted by the thick blue line).