2. The linearized equations of motions
We shall consider the problem of thermal instability in a differentially rotating, incompressible flow stratified in cylindrical shells, and subject to axisymmetric perturbations. The unperturbed configuration, a gaseous star in hydrostatic and thermal equilibrium, is assumed to be axisymmetric, and we investigate the effects of differential rotation on the growing solutions perpendicular to the axis of rotation. For this purpose, we adopt a rotation law of the form , the axis of rotation being confused with the x-axis.
The potentially unstable region is considered to be sufficiently small compared to the stellar radius, rotating between two stable coaxial cylinders, so as to neglect both the Eulerian change in the gravitational acceleration (i.e. the Poisson equation is omitted), as well as the density changes, except for small fluctuations yielding the buoyant force (Boussinesq limit). The linear perturbation counterparts of the coupled Navier-Stokes (with the inertial terms ) and temperature equations then become, in the usual representation (cf. Appendix A; see also Chandrasekhar 1968 for the general procedure),
where is the fluctuation on the temperature, and w are the z-component fluctuations on the vorticity and velocity ; is the fluctuation on the mean molecular weight , and is the fluctuation on the time independent angular frequency . is the kinematic viscosity, the thermometric conductivity ( is the total - thermal plus radiative - conductivity), is the coefficient of thermal expansion, , and is the superadiabatic temperature gradient related to by the well known identity (Cox 1968)
As usual, and are the average temperature gradients inside the perturbed fluid element and outside, in the local surroundings. For adiabatic motions . The efficiency can be expressed in terms of the Peclet number (Maeder 1995). Previous works on linear stability have often deliberately neglected non-adiabatic corrections, in spite of the fact that there resides precisely the physical reason for overstable convection in stars. Actually, the inclusion of radiative losses does not seriously compromise the mathematical tractability of the problem.
The closure relations on µ and are here obtained by imposing a vanishing Lagrangian derivative on both the molecular weight and the specific angular momentum. The condition is a very good approximation, since the effects of the diffusion of elements operate on a much longer timescale than the many other pertinent diffusion timescales in stellar interiors; setting , its linearized form takes the form
The second requirement is more demanding and may not always hold strictly (e.g. magnetic torques); with the help of the mass conservation equation, one finds
which also expresses in cylindrical geometry the absence of meridional circulation. Though very crude and of limited applicability, Eq. 6 leads however to interesting insight on the role of shear as a destabilizing agent.
Eqs. (1)-(3), (5) and (6) are our final set of perturbation equations.
We will next assume normal-mode solutions for all the field variables in the linear stability analysis, and set
where is the growth (or decay) rate of the instability. As noted by Sung (1977), this latter set of eigenfunctions may not be complete for the special problem in hand, and though sufficient conditions for instability remain meaningful, the validity of sufficient conditions for stability is an open question. Also, stellar interiors are inviscid to a very high degree, so let in all our subsequent discussion. By successive elimination one can reduce the system of linearized forms into a single dispersion equation for the linear growth (Appendix B)
where small characters denote real parts, and capital letters the imaginary parts of the cubic coefficients
In the equalities (9) we introduced the (unknown) eddy anisotropy factor , and defined the heat diffusion, dynamical and rotation timescales as the quantities , = and , respectively. Also, we made the following simplification for z sufficiently large
is the difference between the expected tangential velocity due to rotation at a rate , and that obtained by adding the shear velocity field. Then is the time required by two adjacent shells moving at different angular velocities (), to complete relative rotations.
© European Southern Observatory (ESO) 1997
Online publication: May 26, 1998