3.1 Differential rotation of the solar envelope

The internal rotation of the Sun inferred from global helioseismology is shown in Figure 1

. Throughout the convective envelope, the rotation rate decreases monotonically toward the poles by about 30%. Angular velocity contours at mid-latitudes are nearly radial. Near the base of the convection zone, there is a sharp transition between differential rotation in the convective envelope and nearly uniform rotation in the radiative interior. This transition region has become known as the solar tachocline and will be discussed further in the next section (Section 3.2). The rotation rate of the radiative interior is intermediate between the equatorial and polar regions of the convection zone. Thus, the radial angular velocity gradient across the tachocline is positive at low latitudes and negative at high latitudes, crossing zero at a latitude of about $35o$ .

Figure 1:

Angular velocity profile in the solar interior inferred from helioseismology (after Thompson et al., 2003

). In panel (a), a 2D (latitude-radius) rotational inversion is shown based on the subtractive optimally localized averaging (SOLA) technique. In panel (b), the angular velocity is plotted as a function of radius for several selected latitudes, based on both SOLA (symbols, with 1 $s$ error bars) and regularized least squares (RLS; dashed lines) inversion techniques. Dashed lines indicate the base of the convection zone. All inversions are based on data from the Michelson Doppler Imager (MDI) instrument aboard the SOHO spacecraft, averaged over 144 days. Inversions become unreliable close to the rotation axis, represented by white areas in panel (a). Note also that global modes are only sensitive to the rotation component which is symmetric about the equator (courtesy M.J. Thompson & J. Christensen-Dalsgaard).

In addition to the tachocline, there is another layer of comparatively large radial shear in the angular velocity near the top of the convection zone. At low and mid-latitudes there is an increase in the rotation rate immediately below the photosphere which persists down to $r ~ 0.95Ro .$ . The angular velocity variation across this layer is roughly 3% of the mean rotation rate and according to the helioseismic analysis of Corbard and Thompson (2002

) $_O_$ decreases within this layer approximately as $-1 r$ . At higher latitudes the situation is less clear. The radial angular velocity gradient in the subsurface shear layer appears to be smaller and may switch sign (Corbard and Thompson, 2002 ).

Although helioseismic inversions become less reliable at high latitudes (Section 2.1), available data indicate that the monotonic decrease of angular velocity with latitude continues to the polar regions. Moreover, the inferred rotation rate of the polar regions is even slower than that given by a smooth extrapolation of the rotation rate at low and mid-latitudes (Schou, 1998 ). This is a striking result, since flows approaching the rotation axis might be expected to spin up the polar regions if they tend to conserve their angular momentum (cf. Sections 6.3 and 6.4).

Finer structure is also present in the rotational inversions, including “wiggles” in the angular velocity contours and propagating, banded zonal flows known as torsional oscillations (Section 3.3). Zonal jets (localized regions of prograde or retrograde flow) may also be present. Schou (1998 ) reported evidence for a prograde polar jet which can also be seen in the RLS (Regularized Least Squares) inversion results of panel b of Figure 1 (dashed line) at a latitude of 75 $o$ and a radius of $~ 0.95R o .$ . However, some data and analysis techniques spanning the same time interval do not reveal such a jet, so its existence is still questionable (Schou et al., 2002 ). Spatial and temporal variations in the rotation rate are particularly apparent near the poles where the small moment arm, $c = r sin h$ , implies large angular velocity variations even for moderate zonal velocities: $_O_ = vf/c$ . Although many of these fluctuations can likely be attributed to observational and analysis errors, some are statistically significant (Toomre et al., 2000 ).

Global helioseismic inversions such as those shown in Figure 1 can only provide the equatorally-symmetric component of the angular velocity but local helioseismology reveals significant asymmetries, particularly in the torsional oscillations (Haber et al., 2002 ; Basu and Antia, 2003 ; Zhao and Kosovichev, 2004 ).