The spatial evolution of small-amplitude unsteady disturbances of an axisymmetric swirling jet is examined theoretically. The slow axial divergence of the jet mean flow is accounted for by using the method of multiple scales and a consistent solution for both the mean flow and unsteady disturbance is derived. Previous work by Lu & Lele (1999) has considered the leading-order analysis, in which the modal eigenvalues are determined from locally parallel theory, but the key feature of our analysis is the solution of the next-order secularity condition for the axial variation of the wave-envelope amplitude.

The swirling jet profile sustains two types of instability waves: the Kelvin–Helmholtz instability associated with axial shear, and a centrifugal instability which arises due to a decrease in circulation with radial distance. The evolution of the disturbance axial wavenumber and envelope amplitude with downstream distance is calculated. Numerical results show that the growth of the centrifugal mode is significantly curtailed as a result of a rapidly decaying envelope amplitude. The shear instability is significantly more amplified by the addition of swirl.

The general solution for the disturbance envelope amplitude breaks down at so-called turning points. This is found to occur for a series of neutral propagating modes. A rescaling in the vicinity of the turning point shows that the amplitude in this region is governed by a parabolic cylinder equation. The modal amplitude is seen to decay very significantly through this turning point, even though the mode is neutral to leading order.