3.3 Torsional oscillations and other temporal variations in the solar rotation

The rotation rate of the Sun varies on evolutionary timescales; it was once much faster. Here we are concerned with variations on the much shorter dynamical timescales of months, years, and decades. We ask whether the differential rotation profile shown in Figure 1

changes significantly over the course of, for example, the solar activity cycle. The answer is yes. There are two distinct cyclical patterns which have been detected in the solar differential rotation, which we will refer to as torsional oscillations and tachocline oscillations.

The most well-established temporal variations in the solar differential rotation are torsional oscillations, which have been studied using global helioseismology, local helioseismology, and surface Doppler measurements (Howard and LaBonte, 1980 ; Ulrich et al., 1988 ; Howe et al., 2000b ; Vorontsov et al., 2002 ; Haber et al., 2002 ; Basu and Antia, 2003 ; Zhao and Kosovichev, 2004 ). These are alternating bands of faster and slower rotation which propagate with a cyclical period of 11 years. At latitudes below about 42 $o$ , the bands propagate equatorward but at higher latitudes they propagate poleward. The low-latitude bands are about $15o$ wide in latitude and extend from the surface down to $r ~ 0.9R o.$ or deeper, possibly to the base of the convection zone. The high-latitude bands are somewhat wider and deeper, possibly extending to the base of the convection zone (Vorontsov et al., 2002 ). The amplitude of the angular velocity variation is about $2 -5 nHz$ , which is roughly 1% of the mean rotation rate. This corresponds to a zonal flow of about $5 -10 m s-1$ .

The 11-year period of the torsional oscillations strongly suggests that they are associated in some way with the 22-year solar activity cycle. Indeed, surface magnetic activity correlates well with the oscillation patterns, with activity belts tending to lie on the poleward side of the faster-rotating bands at low latitudes, migrating toward the equator together as the activity cycle progresses (Zhao and Kosovichev, 2004 ). Recent results by Beck et al. (2002 ) based on time-distance helioseismology indicate that meridional flows may be diverging out of the activity belts, with equatorward and poleward flows correlating well with the faster and slower bands of the torsional oscillations, respectively. There is some evidence that the zonal bands may get slightly faster at times of peak magnetic activity (Zhao and Kosovichev, 2004 ). Some evidence has also been found for higher-order harmonics in the torsional oscillation signal (Vorontsov et al., 2002 ) and for possibly related non-axisymmetric wave patterns having longitudinal wavenumbers up to $m = 8$ (Ulrich, 2001 ).

The second type of oscillation which has been detected in the differential rotation profile, first reported by Howe et al. (2000a ), is distinct from the torsional oscillation in that it has a shorter period, $~ 1.3 yr$ and it is localized around the tachocline and the lower convection zone. Furthermore, there is currently no evidence for latitudinal propagation. Rather, it appears to be a standing wave straddling the base of the convection zone, with angular velocity variations at $r = 0.63Ro .$ out of phase with those at $r = 0.72Ro .$ . The amplitude of the angular velocity variation is about 3 nHz at the equator and perhaps slightly larger, $~ 4 nHz$ , at a latitude of 60 $o$ . The oscillation may not be strictly periodic; the high-latitude signal in particular appears to be somewhat erratic.

The tachocline oscillation signal is not far from the current sensitivity limits of helioseismic inversions so it is difficult to probe in detail. Basu and Antia (2001 ) find roughly periodic variations similar to those reported by Howe et al. (2000a ) but they argue that the result is not statistically significant. A subsequent analysis by Toomre et al. (2003 ) further made the case that the oscillations are indeed real but they appear to be varying in amplitude as the solar cycle proceeds, first waning then waxing. Further monitoring of these oscillations is needed in order to verify their presence and better understand their origin.

In addition to the torsional and tachocline oscillation patterns, the solar rotation, particularly at high latitudes, undergoes monthly variations on the order of a few percent or less which appear to be more random (Toomre et al., 2000 ). Apart from these small variations, the differential rotation profile appears to be remarkably steady. Surface measurements show little variation for well over a century (Gilman, 1974 ; Schröter, 1985 ; Rüdiger, 1989 ). Still, there is some indication that the low-latitude rotation rate as traced by sunspots may increase by as much as a few percent during periods of minimum and to a lesser extent maximum activity (Howard, 1984 ; Hathaway and Wilson, 1990 ; Javaraiah, 2003 ). Longer-term variations may also be present. Javaraiah (2003 ) has considered rotation data from sunspot groups covering the period from $1879- 2002$ and has found possible evidence for several patterns, including a speedup of the equatorial rotation rate by $~ 0.1%$ in alternate sunspot cycles, accompanyed by an increase in the differential rotation (latitudinal angular velocity gradient) and a greater asymmetry between the northern and southern hemispheres.

Sunspot groups may not be accurate tracers of solar rotation. Small variations in their rotational properties may reflect other physics, such as where they are “anchored” to the plasma. Thus, we are only beginning to explore systematic variations in the solar rotation over time scales of years and decades. High-quality helioseismic inversions have only been available for a single sunspot cycle. Continued monitoring of the internal rotation profile via helioseismology promises to provide new insight into its evolution for many years to come.