Spectral direct numerical simulations (DNS) are carried out for a source-sink flow in an annular cavity between two co-rotating disks. When the Reynolds number based on the forced inflow is increased, a self-sustained crossflow instability of finite amplitude is observed. We show that this nonlinear global mode is made up of a front located at the upstream boundary of the absolutely unstable domain, followed by a saturated spiral mode, and that its properties are in good agreement with results of the local stability theory. This structure is characteristic of the so-called elephant mode of Pier & Huerre (J. Fluid Mech. vol. 435, 2001, p. 145). The global bifurcation is subcritical since only large-amplitude initial perturbations are found to trigger the elephant mode. Small-amplitude perturbations induce a long-lasting transient growth but lead eventually to a damped linear global mode, showing that non-parallel effects counteract the absolute instability and restabilize the flow. A similar linear global stabilization due to non-parallel effects has been found in the case of the flow above a single rotating disk. For the single-disk geometry, the existence of an elephant mode would imply, together with results of Davies & Carpenter (2003) a subcritical global instability, which has not yet been demonstrated. Although the present geometry differs from the single-disk case, the existence of a subcritical global bifurcation is now established, allowing a precise analysis of the transition scenarios. © 2008 Cambridge University Press.