Summary: Dynamo action is considered in a conducting cylindrical annulus surrounded by an insulator. The driving velocity field is assumed to be linear in the axial coordinate and to satisfy the incompressible Navier-Stokes equations. Such flows have recently been shown to exist with no forcing other than the similarity structure. Magnetic field instabilities with the same spatial structure are sought. The kinematic eigenvalue problem is found to have two growing modes for moderate values of the magnetic Reynolds number, \(R_m\). As \(R_m\to\infty\) it is shown that the modes are governed by layers on the outer wall. The growing field saturates in a solution to the nonlinear dynamo problem. Three distinct steady solution families are found and the bifurcation structure is investigated.