Differential Rotation in Stars

The stability of differential rotation in the radiative zones of stars is investigated. For sufficiently large x/ (x is the thermal diffusivity and the kinematic viscosity),it is shown that a necessary condition for stability in regions of homogeneous chemical composition is that the angular momentum per unit mass be an increasing function of distance from the rotation axis. In cylindrical coordinates (i:', `o,z), this condition is given by a( 2 )/a > 0 and 3 /az = 0, where is the angular velocity. The condition is also a sufficient one when applied to axisymmetric perturbations. The stable thermal stratification which exists in the radiative zone cannot prevent the instability since, in stars, the thermal diffusivity is much greater than the kinematic viscosity. The turbulent diffusion of angnlar momentum, which arises when the stability condition is violated, is so rapid that it would appear to preclude the fast rotation of the Sun's interior which has been proposed by Dicke * In the absence of the instability associated with thermal diffusion, i.e,if x = 0, Dicke's solar model is found to be stable. Another means whereby angular momentum might be brought up from the solar interior is by the mechanism of spin-down associated with the formation of an Ekman boundary layer just below the solar convective envelope. The transport of angular momentum, either by spin-down or by turbulent diffusion, would result in the mixing of material below the convective zone of solar type stars if an external torque were applied to the stellar surface. Thus, the depletion of lithium and beryllium would be an inevitable consequence of the loss of a significant fraction of the star's initial angular momentum.