On the Interaction of Convection and Rotation in Stars
Abstract
We use a recent formulation of turbulent convection to study rotation in convective media. The formalism is based on the Boltzmann equation for the distribution function of convective blobs and represents an extension of mixinglength theory. The details of the interactions between blobs are introduced through a model of the collision term. We obtain the stress tensor and other correlation functions of a rotating convective fluid from the second moments of the Boltzmann equation. In steady state, and in the absence of largescale circulation, the Reynolds stress must vanish. We use this condition to determine the equilibrium rotation profile in the equatorial plane of convective stars. Even in the simple case of isotropic scattering of convective blobs, the equilibrium rotation profile is not necessarily solid body rotation; solid body rotation arises only when the scattering is perfectly elastic, whereas in the opposite limit of completely inelastic scattering, the equilibrium profile consists of a uniform distribution of specific angular momentum. If the scattering is allowed to be anisotropic, then we find an even wider range of equilibrium rotation profiles; in particular, we find that for certain choices of the parameters it is possible to have a rising rotation profile, similar to that observed in the equatorial plane of the Sun.
In phenomenological models of the solar rotation, the Reynolds stress is written in a generalized form which includes anisotropic viscosity and Aterms, the latter giving a nonzero shear stress even when there is no velocity shear. Our analysis gives rise to both anisotropic viscosity and Aterms quite naturally and shows that both effects should be generically present. We provide analytical expressions for these effects in terms of the parameters of the interblob scattering function.
We also consider the effect of rotation on the condition for convective instability. According to linear analysis, convection in the equatorial plane is suppressed if the epicyclic frequency exceeds the imaginary part of the BruntVäisälä frequency. However, when the scattering of blobs is included in the analysis, we show that the stability condition is modified, and there can be a new kind of instability even when linear theory predicts stability. This new instability has a zero growth rate when the scattering frequency goes to zero but has a growth time comparable to the convection time when the collision frequency is comparable to the BruntVäisälä frequency. Thus, this instability is a secular instability and possibly a finite amplitude instability, which arises primarily as a result of the scattering of eddies. We discuss the generalization of these results for regions away from the equatorial plane.
Finally, we show that the direction of angular momentum flow can be very different in the linear limit compared to fully developed convection. When the convective perturbations are infinitesimally small, corresponding to the linear limit, we show that the angular momentum flow is in such a direction as to drive the system toward a constant angular momentum configuration. This is in agreement with the result obtained by Ryu & Goodman through linear mode analysis of a Keplerian accretion disk. However, in saturated convection, when the perturbations are fully nonlinear, the angular momentum flow is such as to drive the system toward the equilibrium rotation profiles described above. Thus, the angular momentum flux is outward in the case of fully developed convection in a Keplerian disk. This means that convection is a viable mechanism to generate viscosity in accretion disks.
 Publication:

The Astrophysical Journal
 Pub Date:
 November 1995
 DOI:
 10.1086/176409
 Bibcode:
 1995ApJ...453..480K
 Keywords:

 CONVECTION;
 STARS: ROTATION;
 SUN: INTERIOR;
 TURBULENCE;
 SUN: ROTATION