This paper presents an analysis of the Dudley-James two-vortex flow, which inspired several laboratory-scale liquid-metal experiments, in order to better demonstrate its relation to astrophysical dynamos. A coordinate transformation splits the flow into components that are axisymmetric and nonaxisymmetric relative to the induced magnetic dipole moment. The reformulation gives the flow the same dynamo ingredients as are present in more complicated convection-driven dynamo simulations. These ingredients are currents driven by the mean flow and currents driven by correlations between fluctuations in the flow and fluctuations in the magnetic field. The simple model allows us to isolate the dynamics of the growing eigenvector and trace them back to individual three-wave couplings between the magnetic field and the flow. This simple model demonstrates the necessity of poloidal advection in sustaining the dynamo and points to the effect of large-scale flow fluctuations in exciting a dynamo magnetic field.