CONVECTIVE INSTABILITY IN DIFFERENTIALLY ROTATING-DISKS

Abstract

We perform a normal mode analysis for nonaxisymmetric perturbations in a thin, differentially rotating disk with a vertical structure that is isothermal and convectively unstable. The vertical gravity is assumed to be external and constant. The perturbation scale is assumed to be much shorter than the radius of the disk but comparable to or less than the thickness. The initial value problem is formulated in shearing coordinates. Dispersion relations are obtained for the three limiting cases of zero shear, axisymmetric perturbations, and small radial wavelengths. The full effects of shear are studied by integrating numerically the initial value problem. The main results of the paper are as follows: (1) Nonaxisymmetric local Fourier modes have a radial wavenumber that increases linearly with time in proportion to the shear times the azimuthal wavenumber. (2) While Coriolis forces exert stabilizing effects on the convective modes, reducing their growth rate and the range of unstable wavelengths, shear has destablizing effects inasmuch as it reduces the epicyclic frequency at a given angular velocity. (3) In a Keplerian disk, perturbations with azimuthal wavelengths about 2 times smaller than vertical wavelengths grow exponentially. Otherwise, perturbations do not grow until radial wavelengths become several times smaller than vertical wavelengths but grow exponentially after that. The exponential growth rate is roughly the limiting growth rate-the absolute value of the imaginary Brunt-Vaisala frequency-at infinitely small radial wavelengths. (4) A dispersion relation which can be used to estimate the instantaneous growth rates, or at least, the power bound of the instantaneous growth rates for nonaxisymmetric perturbations in differentially rotating disks is presented. (5) The angular momentum flux of the linear modes is nonzero only for nonaxisymmetric disturbances, and for these the flux is predominantly inward, i.e., in the direction of increasing angular velocity