Instabilities in two-dimensional spatially periodic flows. Part II: Square eddy lattice

A numerical study is performed on the linear stability of square arrays of alternating vortices with emphasis on confinement effects and linear friction. The appearance of a global rotation as a result of instability in a strongly confined system consisting of four vortices is explained in terms of the first unstable mode and the calculated critical parameter is found to be in agreement with magnetohydrodynamic experiments. The existence of bicritical points in systems with more than four vortices is demonstrated. Stability of the unbounded system is treated in the framework of Floquet theory. It is found that, in the presence of linear friction, the basic flow is unstable with respect to quasiperiodic perturbations with generally incommensurable spatial frequencies. This lays the foundation for a hypothesis about transition to turbulence via spatial quasiperiodicity.