Convective versus absolute instability in flow between counterrotating cylinders.

Abstract

The linear stability of flow in a Couette-Taylor system with counterrotating cylinders is posed as an initial-value problem. The time-asymptotic solution is found by the method of steepest descent. We identify the saddle points of the complex dispersion relation and, from the growth rates evaluated at the saddle points, determine a Reynolds number range ${\mathit{R}}_{\mathit{c}}$\ensuremath{\le}R\ensuremath{\le}${\mathit{R}}_{\mathit{a}}$ within which the instability is convective and beyond which (Rg${\mathit{R}}_{\mathit{a}}$) it is absolute. The amount by which ${\mathit{R}}_{\mathit{a}}$ exceeds ${\mathit{R}}_{\mathit{c}}$ can be of order 1%, but loci in parameter space of zero group velocity are found where the instability is absolute at onset (${\mathit{R}}_{\mathit{a}}$=${\mathit{R}}_{\mathit{c}}$). Bicritical curves separating regions with different azimuthal wave numbers are compared for convective and absolute instabilities. The analysis reveals a new type of bicriticality where two different axial wave numbers simultaneously become unstable.