6.2 Convection structure

Figure 9

illustrates the variety of convective patterns which have been found in high-resolution simulations of global solar convection. Three simulations are shown, including Case M3 of Brun et al. (2004

), Case F of Brun et al. (2005 ) and Case D2 of DeRosa et al. (2002

). The primary difference between Cases M3 and F is that the latter is less dissipative and therefore more turbulent (higher Rayleigh and Reynolds numbers). Case D2 is comparable to Case F but with a thin-shell geometry; the lower boundary was set at $r1 = 0.92Ro .$ as opposed to $r1 = 0.62Ro .$ . Cases F and D2 both have an upper boundary at $r = 0.98R 2 o .$ , somewhat closer to the solar photosphere than Case M3 ( $r2 = 0.96Ro .$ ). Case M3 is the only one of the three which includes magnetism, although this does not have a substantial influence on the convective patterns shown in Figure 9

(Brun et al., 2004

At intermediate Rayleigh (and Reynolds) numbers, there is a marked contrast between the convective structure at low and high latitudes (Figure 9 , panel a). Near the equator, the convection is dominated by extended downflow lanes oriented north-south which propagate in longitude faster than the local differential rotation (Miesch et al., 2000 ). These are reminiscent of the banana cells in earlier more laminar simulations (see Section 6.1) but they are not strictly periodic in longitude, they extend only to mid-latitudes, and they are asymmetric with respect to upflow and downflow due to the density stratification (cf. Section 5.2). Near the poles the convection patterns are more isotropic and homogeneous and the characteristic spatial scales are somewhat smaller.

This variation in convective patterns results arises from the influence of rotation and some insight into its origin can be gained from linear theory (Busse, 1970 ; Gilman, 1975 ; Busse and Cuong, 1977 ). In order to minimize the stabilizing influence of the Coriolis force, convection at low latitudes tends to favor flows which are perpendicular to the rotation axis. If the rotation is rapid, columnar convection cells are preferred which align with the rotation axis and propagate in a prograde direction due to their tendency to conserve angular momentum (or potential vorticity) under the influence of the spherical geometry and density stratification; in this sense they may be regarded as thermal Rossby waves (Glatzmaier and Gilman, 1981 ; Busse, 2002 ). At high latitudes, inside the tangent cylinder⁴ overturning motions can no longer remain perpendicular to $_O_0$ , resulting in more isotropic cells with smaller horizontal scales. In more turbulent parameter regimes (higher Rayleigh and Reynolds numbers), the convection near the top of the domain exhibits a more granulation-like character across the shell as shown in panel b of Figure 9 . As in simulations of turbulent compressible convection in Cartesian domains (see Section 5.2), the convection structure is dominated by an intricate, interconnected network of downflow lanes amidst broader, weaker upflows. Although the patterns appear relatively homogeneous and isotropic with little indication of banana cells, broad upwellings and extended north-south lanes still occur at low latitudes within the more intricate downflow network. These extended downflow lanes generally penetrate deeper into the convection zone than the smaller-scale network patterns (see Figure 12 ) and play an important role in maintaining the differential rotation (see Section 6.3). Horizontal rolls analogous to north-south downflow lanes are present in even the most turbulent Cartesian simulations of turbulent compressible convection when the rotation vector is made horizontal in order to simulate the equatorial regions (Brummell et al., 2002b ).

Although these convective patterns are reminiscent of granulation or supergranulation, their scale is much larger. By eye, the predominant convective cells in panel b of Figure 9 appear to span roughly 10 angular degrees, which corresponds to a horizontal scale of $120 Mm$ . More localized, swirling structures are also evident near the interstices of the downflow network at mid-latitudes. The power spectrum of the radial velocity field peaks at spherical harmonic wavenumbers of $l ~ 50- 60$ , which corresponds to $~ 80 Mm$ . Recall that the characteristic scales of granulation and super-granulation are about $1- 2 Mm$ and $30 Mm$ , respectively (Section 2.3).

Convective motions at supergranular scales have been reported in global simulations by DeRosa et al. (2002 ) who focused on the upper regions of the solar convection zone. Higher spatial resolution was achieved by limiting the simulation domain to radii between $0.92- 0.98Ro .$ and by imposing a four-fold periodicity in longitude. The convection structure in one of these simulations is illustrated in panel c of Figure 9 . The pattern exhibits a hierarchy of scales, from supergranular-scale mottling to a network of larger cells and extended north-south downflow lanes more comparable to the deep-shell simulations (cf. Figure 9 , panel b). Although provocative, it is premature to identify this small-scale convection pattern too closely with supergranulation on the Sun. Solar supergranulation may involve dynamics which are not captured in these global simulations such as ionization effects or self-organization processes involving smaller-scale granules (Rast, 2003 ). On the other hand, although simulations of granulation with large aspect ratios exhibit structure on mesogranule scales ( $~ 5 Mm$ ), they have not yet achieved larger-scale patterns so the origin of supergranulation remains unclear (Rincon et al., 2005 ; see also Simon and Weiss , 1991 ).

Figure 10:

Movie showing the temporal evolution of the radial velocity near the top of the shell ( $r = 0.98Ro .$ ) in Case F is shown in an orthographic projection as in Figure 9

. The movie covers a time span of 7 days.

In turbulent parameter regimes, the downflow network evolves rapidly, changing substantially over the course of a few days (recall that the rotation period is about a month). This is demonstrated in Figure 10

which follows the radial velocity field near the top of the convection zone in Case F. Advection and distortion of the downflow network by the differential rotation is evident, with low-latitude patterns moving eastward and high-latitude patterns moving westward relative to the rotating coordinate system. Downflow lanes continually merge and re-form as upwellings diverge and fragment. Particularly at high and mid-latitudes, numerous localized vortices appear and disappear near the interstices of the downflow network, often forming new upwellings via the centrifugal siphoning of fluid from below (Brandenburg et al., 1996

; Brummell et al., 1996

; Miesch, 2000

). Such vortices are fed by converging horizontal flows which tend to conserve their angular momentum, spinning up in a cyclonic sense⁵ due to the Coriolis force. This results in intense, intermittent downflow plumes spinning with cyclonic vorticity. These vortical downflow plumes appear as cool spots in the temperature field as shown in panel a of Figure 11

. Global temperature variations are also apparent, with equatorial and polar regions a few K warmer than mid-latitudes. The local maxima at the poles are often more pronounced in the entropy field than the temperature field, which has implications for the thermal wind component of the differential rotation (Section 6.3). The downflow network is also faintly visible in the temperature field of panel a in Figure 11

; in many simulations it leaves a more noticeable imprint (e.g., Thompson et al., 2003 , Figure 13).

Figure 11:

The temperature (a), radial vorticity (b), and horizontal divergence (c) near the top of the convection zone in Case F. The time instance and projection are as in Figure 9

The cyclonic nature of the downflow lanes and plumes is evident in the radial vorticity field, shown in panel b of Figure 11

. This pattern stands out amid a background of weaker anti-cyclonic vorticity associated with diverging upflows. As expected, the horizontal divergence field, shown in panel c of Figure 11

correlates well with the vertical velocity field shown in panel b of Figure 9

. The relative magnitudes of the vortical and divergent components of the horizontal velocity field near the top of the convection zone can potentially be a point of contact between numerical simulations and helioseismic observations Section 3.5. In simulations, the two are generally comparable (the rms values of the vertical vorticity and horizontal divergence fields shown in Figure 11

are $1.6 × 10-5 s-1$ and $1.5 × 10-5 s-1$ , respectively).

Figure 12:

The radial velocity (a), temperature (b), and enstrophy (c) are shown for Case F in the mid convection zone. The time instance and projection are as in Figure 9

Deeper in the convection zone, the flow structure changes dramatically as illustrated in Figure 12

. Only the strongest downflow plumes and lanes in the near-surface network penetrate to the mid convection zone and the network loses its connectivity. Cool, vortical, intermittent plumes dominate but coherent north-south downflow lanes still persist at low latitudes. In the near-surface layers, the enstrophy (vorticity squared) is dominated by the intense cyclonic vertical vorticity found in the downflow network (Figure 11

, panel b). In the mid convection zone, enstrophy is still concentrated in downflows but is now dominated by horizontal entrainment vortices, forming rolls and ’smoke rings’ near the periphery of lanes and plumes (Figure 12

, panel c).