Prof Steven Balbus FRS, FInstP

Abstract:

We study the local stability of stratified, differentially rotating fluids to axisymmetric perturbations in the presence of a weak magnetic field and of finite resistivity, viscosity, and heat conductivity. This is a generalization of the Goldreich-Schubert-Fricke (GSF) double-diffusive analysis to the magnetized and resistive, triple-diffusive case. Our fifth-order dispersion relation admits a novel branch that describes a magnetized version of multi-diffusive modes. We derive necessary conditions for axisymmetric stability in the inviscid and perfect-conductor (double-diffusive) limits. In each case, rotation must be constant on cylinders and angular velocity must not decrease with distance from the rotation axis for stability, irrespective of the relative strength of viscous, resistive, and heat diffusion. Therefore, in both double-diffusive limits, solid-body rotation marginally satisfies our stability criteria. The role of weak magnetic fields is essential to reach these conclusions. The triple-diffusive situation is more complex, and its stability criteria are not easily stated. Numerical analysis of our general dispersion relation confirms our analytic double-diffusive criteria but also shows that an unstable double-diffusive situation can be significantly stabilized by the addition of a third, ostensibly weaker, diffusion process. We describe a numerical application to the Sun's upper radiative zone and establish that it would be subject to unstable multidiffusive modes if moderate or strong radial gradients of angular velocity were present.

Abstract:

Just as correlations between fluctuating radial and azimuthal velocities produce a coherent stress contributing to the angular momentum transport in turbulent accretion disks, correlations in the velocity and temperature fluctuations produce a coherent energy flux. This nonadvective thermal energy flux is always of secondary importance in thin radiative disks, but cannot be neglected in nonradiative flows, in which it completes the mean field description of turbulence. It is nevertheless generally ignored in accretion flow theory, with the exception of models explicitly driven by thermal convection, for which it is modeled phenomenologically. This flux embodies both turbulent thermal convection and wave transport, and its presence is essential for a proper formulation of energy conservation, whether convection is present or not. The sign of the thermal flux is likely to be outward in real systems, but the restrictive assumptions used in numerical simulations may lead to inward thermal transport, in which case qualitatively new effects may be exhibited. We find, for example, that a static solution would require inward, not outward, thermal transport. Even if it were present, thermal convection would be unlikely to stifle accretion but would simply add to the outward rotational energy flux that must already be present.