3.2 The tachocline

The tachocline is a transition layer between two distinct rotational regimes: the differentially-rotating solar envelope and the radiative interior where the rotation is uniform, within the error estimates of the inversions (Figure 1

). This transition is sharp and it occurs near the base of the convection zone as determined by helioseismic inversions and solar models (Section 3.6), implying that convection is responsible for the differential rotation of the envelope (Section 4.3). Although some authors incorporate structural information (e.g., subadiabaticity), most define the tachocline solely by means of the rotation profile. We will follow the latter convention.

Recent helioseismic estimates by Charbonneau et al. (1999a ) and Basu and Antia (2003 ) indicate that the tachocline is centered at $rt ~ 0.693 ± 0.003Ro .$ near the equator. This is below the convection zone base of $rb = 0.713 ± 0.003Ro .$ but it may lie within the overshoot region (Section 3.6). At higher latitudes, the location of the tachocline shifts upward, reaching $r ~ 0.717 ± 0.003R t o .$ at a latitude of 60 $o$ (Charbonneau et al., 1999a ; Basu and Antia, 2003 ). Thus, the tachocline is significantly prolate. This is in contrast to the base of the convection zone, $rb$ , in which helioseismic inversions have not yet detected any significant latitudinal variation (Section 3.6).

Estimates of the width of the tachocline vary according to how it is defined. Charbonneau et al. (1999a ) characterize the transition in terms of an error function

and then estimate the best-fit parameters using several inversion techniques. Their results yield a tachocline thickness of $Dt/Ro. = 0.039 ± 0.013$ at the equator and $Dt/Ro. = 0.042 ± 0.013$ at a latitude of $60o$ , suggesting that the tachocline may get somewhat wider at high latitudes but that the result is not statistically significant. On the other hand, Basu and Antia (2003

) argue for a statistically significant increase in the tachocline thickness with latitude, from $Dt ~ 0.016Ro .$ at the equator to $Dt ~ 0.038Ro .$ at latitudes of $o 60$ (when the width is defined as in Charbonneau et al., 1999a ). Furthermore, they suggest that the variation may not be smooth; there may be a sharp transition from a narrow tachocline at low latitudes to a wider tachocline at high latitudes, possibly associated with the sign of the radial angular velocity gradient which reverses at a latitude of $~ 35o$ . Other estimates for the width of the tachocline range from $0.01Ro .$ to $0.09Ro .$ (Kosovichev, 1996 ; Basu, 1997

; Corbard et al., 1999 ; Elliott and Gough, 1999 ; Basu and Antia, 2001

These helioseismic results suggest that the tachocline lies almost entirely below the convective envelope at low latitudes but it may extend well into the convection zone at high latitudes. Moreover, it appears that the tachocline contains the overshoot region but extends beyond it, perhaps both above and below. However, these results may need to be reexamined in light of new determinations of elemental abundances in the solar envelope, which has important implications for helioseismic inversions (Asplund et al., 2005 ; Bahcall et al., 2005 ).

Throughout most of the tachocline, the vertical shear in the mean zonal velocity almost an order of magnitude larger than the latitudinal shear: $dvf/dr ~ ± 1.5 × 10- 6 s-1$ whereas $r- 1dvf/dh ~ 2 × 10- 7 s-1$ . The exception is at latitudes of $~$ 35 $o$ where $dvf/dr$ changes sign. The total change in the zonal velocity across the tachocline is about $100 m s- 1$ at the equator and somewhat less at high latitudes, $-1 ~ 90 m s$ .