We present calculations of optimal linear growth in the Batchelor (or q) vortex. The level of transient growth is used to quantify the effect of the viscous centre modes found at large Reynolds number and large swirl. The viscous modes compete with inviscid-type transients, which are seen to provide faster growth at short times. Following a smooth transition, the viscous modes emerge as dominant in a different regime at later times. A comparison is drawn with two-dimensional shear flows, such as boundary layers, in which weak instability modes (Tollmien–Schlichting waves) also compete with inviscid transients (streamwise streaks). We find the competition to be more evenly balanced in the Batchelor vortex, because the inviscid transients are damped faster in a swirling jet than a two-dimensional shear flow, so that despite their weak growth rates the viscous modes may be relevant in some situations. The elliptic instability of a Batchelor vortex subject to a stationary strain field is considered by theoretical and numerical means in the regime of large Reynolds number and small axial flow. In the theory, the elliptic instability is described as a resonant coupling of two quasi-neutral normal modes (Kelvin modes) of the Batchelor vortex of azimuthal wavenumbers m and m + 2 with the underlying strain field. The growth rate associated with these resonances is computed for different values of the azimuthal wavenumbers as the axial flow parameter is varied. We demonstrate that the resonant Kelvin modes m = 1 and m = −1 which are the most unstable in the absence of axial flow become damped as the axial flow is increased. This is shown to be due to the appearance of a critical layer which damps one of the resonant Kelvin modes. However, the elliptic instability does not disappear. Other combinations of Kelvin modes m = −2 and m = 0, then m = −3 and m = −1 are shown to become progressively unstable for increasing axial flow. A complete instability diagram is obtained as a function of the axial flow parameter for several values of the strain rate and Reynolds number.The numerical study considers a system of two counter-rotating Batchelor vortices in which the strain field felt by each vortex is due to the other vortex. The characteristics of the most unstable linear modes developing on the frozen base flow are computed by direct numerical simulations for two axial flow parameters and compared to the theory. In both cases, a very good agreement is obtained for the most unstable modes. Less unstable modes are also identified in the numerics and shown to correspond to peculiar resonances involving Kelvin modes from branches of different labels.