Angular Momentum Transport in Magnetized Stellar Radiative Zones. IV. Ferraro’s Theorem and the Solar Tachocline

Abstract

We consider the circumstances under which the latitudinal di erential rotation of the solar convective envelope can (or cannot) be imprinted on the underlying radiative core through the agency of a hypo-thetical weak, large-scale poloidal magnetic Ðeld threading the solar radiative interior. We do so by con-structing steady, two-dimensional axisymmetric solutions to the coupled momentum and induction equations under the assumption of a purely zonal Ñow and time-independent poloidal magnetic Ðeld. Our results show that the structure of the interior solutions is entirely determined by the boundary con-ditions imposed at the core-envelope interface. SpeciÐcally, in the high Reynolds number regime a poloi-dal Ðeld having a nonzero component normal to the core-envelope interface can lead to the transmission of signiÐcant di erential rotation into the radiative interior. In contrast, for a poloidal Ðeld that is con-tained entirely within the radiative core, any di erential rotation is conÐned to a thin magnetoviscous boundary layer located immediately beneath the interface, as well as along the rotation/magnetic axis. We argue that a magnetically decoupled conÐguration is more likely to be realized in the solar interior. Consequently, the helioseismically inferred lack of di erential rotation in the radiative core does not nec-essarily preclude the existence of a weak, large-scale poloidal Ðeld therein. We suggest that such a Ðeld may well be dynamically signiÐcant in determining the structure of the solar tachocline. Subject headings : Sun : interior È Sun : oscillations È Sun : rotation 1. INTRODUCTION Measurements of solar p-mode oscillation frequencies have been used to infer the angular velocity of rotation)(r, h) throughout much of the SunÏs interior. Numerous analyses of helioseismic observations reveal that the photo-spheric dependence of) on h also describes the variation of the internal rotation rate with latitude at any radius r within the convective envelope, R CE (B 0.713 R _) ¹ r ¹ (Brown et al. 1989 ; Tomczyk, Schou, & Thompson R _ 1995). Somewhat below the base of the convection zone, there exists a thin shear layer (called the tachocline by Spiegel & Zahn 1992), within which the surface-like rota-tion of the convection zone changes to near-uniform rota-tion in the radiative interior. The central radius of the tachocline appears to lie within the range 0.695È0.705 R _ , and its thickness is estimated to be (Kosovichev [0.05 R _ 1996 ; Basu 1997 ; Charbonneau et al. 1998a ; Corbard et al. 1999). In the radiative layers beneath the tachocline,) is almost independent of both r and h ; in fact, the region 0.1 seems to rotate nearly rigidly at a rate like R _ [ r [ 0.7 R _ that of the solar surface at a latitude of about 30¡ (Charbonneau et al. 1998b). The apparent absence of signiÐcant di erential rotation in the radiative core of the Sun represents a stringent con-straint on potential mechanisms for internal angular momentum redistribution. Evidently, the process(es) at work in this region must be able to counteract the e ects of the shearing stress exerted by the convective envelope on the stable layers just beneath it. Models for the solar rota-tional evolution in which the internal transport of angular