Stability of periodic arrays of vortices

Abstract

The stability of periodic arrays of Mallier-Maslowe or Kelvin-Stuart vortices is discussed. We derive with the energy-Casimir stability method the nonlinear stability of this solution in the inviscid case as a function of the solution parameters and of the domain size. We exhibit the maximum size of the domain for which the vortex street is stable. By adapting a numerical time-stepping code, we calculate the linear stability of the Mallier-Maslowe solution in the presence of viscosity and compensating forcing. Finally, the results are discussed and compared to a recent experiment in fluids performed by Tabeling et al.~[Europhysics Letters {\bf 3}, 459 (1987)]. Electromagnetically driven counter-rotating vortices are unstable above a critical electric current, and give way to co-rotating vortices. The importance of the friction at the bottom of the experimental apparatus is also discussed.