Long's vortex revisited

We reconsider exact solutions to the Navier-Stokes equations that describe a vortex in a viscous, incompressible fluid. This type of solution was first introduced by Long (J. Atmos. Sci., vol. 15 (1), 1958, p. 108) and is parameterized by an inverse Reynolds number ε. Long's attention (and that of many subsequent investigators) was centred upon the quasi-cylindrical (QC) case corresponding to ε= 0. We show that the limit ε→0 is not straightforward, and that it reveals other solutions to this fundamental exact reduction of the NavierStokes system (which are not of QC form). Through careful numerical investigation, supported by asymptotic descriptions, we identify new solutions and describe the full parameter space that is spanned by and the pressure at the vortex core. Some erroneous results that exist in the literature are corrected. © 2009 Cambridge University Press.