Colloquium: Unusual dynamics of convection in the Sun

The Sun is our nearest star; it is also the most important star that determines life on Earth. A large variety of phenomena observed on the Sun’s surface, with potential impact on Earth, is thought to arise from turbulent convection in Sun’s interior, this being the dominant mode of heat transport within the outer envelope at $r\gtrsim 0.715R_{\odot }$. However, convection in the Sun differs in most of its aspects from convection processes known on Earth, certainly those under controlled laboratory conditions, thus seriously challenging existing physical models of convective turbulence and boundary conditions in the Sun. Solar convection is a multiscale-multiphysics phenomenon including the transport of mass, momentum, and heat in the presence of rotation, dynamo action, radiation fluxes, and partial changes in chemical composition. Standard variables of state such as pressure, mass density, and temperature vary over several orders of magnitude within the convection region, thus introducing immense stratification. Although the Sun has been explored intensely, observational evidence on the structure and intensity of turbulent convection processes remains indirect and essentially limited to observations of the granular convection patterns at the surface and helioseismologic data that probe the propagation of sound waves in the interior. In this Colloquium characteristic scales and dimensionless parameters are discussed, particularly from the perspective of laboratory convection, a research field that has progressed significantly in the last few decades. The estimates and calculations of solar conditions given here are based mostly on the standard solar model S of Christensen-Dalsgaard et al., which is a mean field model of solar convection. Light is shed on existing results to gain a deeper understanding of dynamical aspects of solar convection.

Figure 1

Solar convection patterns at different scales. (a) Granulation. The characteristic length, time, and velocity scales of the granules are, respectively, ${\ell }_{\mathrm{G}}\approx 1000\mathrm{km}$, ${\tau }_{\mathrm{G}}\approx 10\min$, and $v_{\mathrm{G}}\approx 3\mathrm{km}/\mathrm{s}$. More details on granulation can be found, for example, in [1] and [177]. (b) Supergranulation indicated by horizontal surface flow divergences. The characteristic length, time, and velocity scales are, respectively, ${\ell }_{\mathrm{SG}}\approx 3\times {10}^4\mathrm{km}$, ${\tau }_{\mathrm{SG}}\approx 24\hspace{0.17em}\hspace{0.17em}\text{h}$, and $v_{\mathrm{SG}}\approx 500\mathrm{m}/\mathrm{s}$; see also [180]. [128] shared similar images. (c) Giant cells. The characteristic length, time, and velocity scales are estimated to be ${\ell }_{\mathrm{GC}}\sim H\sim 2\times {10}^8\mathrm{m}$, ${\tau }_{\mathrm{GC}}\approx 1\hspace{0.17em}\hspace{0.17em}\text{month}$, and $v_{\mathrm{GC}}\sim 10\mathrm{m}/\mathrm{s}$, respectively ([91]). Their interpretation is still in question.

Figure 2

Radial dependence of (a) temperature $T$, (b) mass density $\rho$, and (c) pressure $p$ across the CZ from model S. The temperature drops down to 5779 K at the solar surface. The vertical dashed lines indicate the depth to which helioseismology can access fluctuations in velocity. (d) Generally accepted abundances of hydrogen ($X$), helium ($Y$), and all heavier elements ($Z$). The thin solid lines indicate the typical values of $X=0.74$ and $Y=0.24$ in the CZ. (e) Scale heights of temperature, pressure, and density ${\mathcal{H}}_T$, ${\mathcal{H}}_p$, and ${\mathcal{H}}_{\rho }$. (f) Plot of the acceleration due to gravity. We indicate the extent of the CZ in all six panels by a shaded background, which starts at $r_{*}\approx 0.715R_{\odot }$.

Figure 3

Radial variation of the different dimensionless temperature derivatives ${\nabla }_m$. Data are calculated from model S. (a) ${\nabla }_{\gamma }$, ${\nabla }_s$, and $\nabla$ are shown across the entire convection zone (CZ), indicated as a shaded background. (b) Enlargement close to the solar surface. Depth $z$ from the surface is given in km. The range where ${\nabla }_s>\nabla$ is darker shaded. Line styles are the same as in (a). (Inset) Superadiabaticity $ϵ=\nabla -{\nabla }_s$ across the CZ.

Figure 4

(a) Opacity profile ${\mathcal{K}}_{\gamma }(r)$ in the solar convection zone. (b) Transport coefficients across the solar convection zone. The temperature diffusivity ${\kappa }_{\gamma }(r)$ in ${\mathrm{m}}^2/\mathrm{s}$, the dynamic viscosity ${\eta }_i(r)$ in $\mathrm{kg}/(\mathrm{m}\mathrm{s})$, and the magnetic diffusivity ${\lambda }_e(r)={[{\mu }_0{\sigma }_e(r)]}^{-1}$ in ${\mathrm{m}}^2/\mathrm{s}$ are shown. (c) Variation of the thermal and magnetic Prandtl numbers across the solar convection zone. The shaded area to the right stands for $r_i\gtrsim 0.98R_{\odot }$, where the hydrogen of the solar plasma is partially ionized. The vertical dotted lines in all panels mark $r=r_{*}$.

Figure 5

Effective temperature difference $\mathrm{\Theta }(r)$ in kelvin that drives convective motion in the solar CZ. The quantity is calculated by Eq. (30). (a) Full profile across the entire CZ. (b) Enlargement of the bottom of the CZ. (c) Enlargement of the surface layer. Both magnifications are indicated as dotted boxes in (a). The shaded area to the right in (a) stands again for $r\gtrsim 0.98R_{\odot }$, where the hydrogen of the solar plasma is only partially ionized. The vertical dashed black lines in (a) and (b) mark $r=r_{*}$. The intersection point of the two dashed lines in (c) is an estimate for a thermal boundary layer thickness by means of the slope method; see Sec. 7a.

Figure 6

Variation of Rayleigh, Reynolds, Rossby, and Chandrasehkar numbers across the solar convection zone based on model S. (a) The profile of the Rayleigh number ${\text{Ra}}_{\odot }(r)$. Free-fall velocity and Rayleigh numbers are calculated with respect to the pressure scale height. (Inset) Variation of the free-fall velocity $U_f$ [see Eq. (31)] and the adiabatic speed of sound $c_s$ [see Eq. (27)]. (b) Variation of flow and magnetic Reynolds numbers across the CZ. Also shown is the pressure scale height in meters. (c) Variation of Rossby and Chandrasekhar numbers across the CZ. The shaded area to the right in both figures corresponds to $r_i\gtrsim 0.98R_{\odot }$ where the hydrogen of the solar plasma is only partially ionized. In (c), we add the Chandrasekhar limit ${\mathrm{Q}}_{c,\odot }={\text{Ra}}_{\odot }/{\pi }^2$; see Sec. 5.

Figure 7

Interaction and coexistence of physical processes with turbulent convection regimes in the solar convection zone. A depth of $z=2\times {10}^3\mathrm{km}$ corresponds to $r\approx 0.997R_{\odot }$, and $z=1.4\times {10}^4\mathrm{km}$ corresponds to $r\approx 0.98R_{\odot }$. At the top and bottom differential rotation is large in magnitude compared to that in the bulk.

Figure 8

Numerical simulations of solar convection. (a) Schematic of the solar convection zone with highlighted surface layer (outer thick line) and tachocline (inner thick line). (b) Simulations of surface convection from [177]. Left panel: vertical (line of sight) velocity component $v_{\mathrm{LOS}}$. Right panel: mean circular polarization $⟨p_{\mathrm{circ}}⟩$ of the Fe I line at $5250.2\hspace{0.17em}\hspace{0.17em}\AA$ as a measure of the magnetic field strength. One arc second corresponds to 727 km. (c) Radial velocity at (top panel) $r=0.98R_{\odot }$ and (bottom panel) $r=0.92R_{\odot }$ showing upflows and downflows by [142]. (d) Penetrative convection into the tachocline by [97]. The figure shows contours of the vertical velocity component in a vertical cross section. The white line marks the border between the convection and radiation zones.

Figure 9

Snapshots of the vertical velocity component $u_z/U_f$ from direct numerical simulations of Rayleigh-Bénard convection (RBC) at $\mathrm{Ra}={10}^5$ in a large-aspect-ratio closed cell with $L/H=25$. Length $L$ is the extension of the square cross section. (a) Prandtl number $\mathrm{Pr}=0.001$, which results in a bulk Reynolds number $\mathrm{Re}=u_{\mathrm{rms}}\sqrt{\mathrm{Ra}/\mathrm{Pr}}/U_f=4810$. The color scale from blue to red covers $u_z/U_f\in [-0.8,0.8]$. The three-dimensional computations in (a) and (d) were conducted with a grid resolution of ${9600}^2\times 640$ points, applying a code by [122]. (b) Convection in air with $\mathrm{Pr}=0.7$ results to $\mathrm{Re}=92$. Here $u_z/U_f\in [-0.4,0.4]$. (c) Convection in water at $\mathrm{Pr}=7$. This RBC flow results in $\mathrm{Re}=11$. Here $u_z/U_f\in [-0.2,0.2]$. (d)–(f) Magnifications of (a)–(c). Data of the two larger Prandtl numbers are from [164]. (g) Corresponding mean temperature profiles at $\mathrm{Ra}={10}^5$ as a function of $\mathrm{Pr}$ becoming nearly linear for $\mathrm{Pr}=0.001$. (h) Eddy diffusivity profiles ${\kappa }_e(z)$ for the three lowest corresponding Prandtl numbers.

Figure 10

Reynolds number vs Rayleigh number at $\mathrm{Pr}=0.005$ (squares), $\mathrm{Pr}=0.021$ (circles), and $\mathrm{Pr}=0.7$ (triangles) with corresponding scaling laws. Open symbols are extrapolations to $\mathrm{Ra}={10}^{10}$. Data and fits are from [189]. Inset: extrapolated and measured data at $\mathrm{Ra}={10}^{10}$ vs Pr. The solid line stands for $\mathrm{Re}=1.7\times {10}^4{\mathrm{Pr}}^{-0.6}$. Asterisks are from [155].

Figure 11

Streamlines of (a) instantaneous and (b) time-averaged velocity fields viewed from the top. The averaging time in (b) is about $60H/U_f$. DNS data for $\mathrm{Ra}={10}^5$, $\mathrm{Pr}=0.005$, and aspect ratio $\mathrm{\Gamma }=25$ are from [164], who applied a spectral element method ([188]).