We examine non-linear spiral flow in the Taylor–Couette problem for a wide gap with axially periodic conditions. We present a highly efficient computational method adapted to this problem, based on continuation methods applied to a pseudospectral discretization of the Navier–Stokes equations in a rotating frame of reference. The spiral flow is computed in a wide range of parameters, and different features are explored in detail: domain of existence of the flow, behavior for high Reynolds number, appearance of axial flows, dependency on parameters, and stability against helical disturbances. A first integral is obtained and used to describe the particle trajectories in the fluid. This description shows that the axial and radial motion of the particles is mainly confined within an internal boundary layer. ©1998 American Institute of Physics.