Symmetry-Breaking Bifurcations in Two-Dimensional Square Vortex Flows

We present a theoretical study of spatial symmetries and bifurcations in a laterally bounded two-dimensional flow composed of approximately square vortices. The numerical setting simulates a laboratory experiment wherein a shallow electrolyte layer is driven by a plane-parallel force that is nearly sinusoidal in both extended directions. Choosing an integer or half-integer number of forcing wavelengths along each direction, we generate square vortex flows invariant under different spatial symmetries.  We then map out the sequence of symmetry-breaking bifurcations leading to the formation of fully asymmetric flows. Our analysis reveals a gallery of pitchfork and Hopf bifurcations, both supercritical and subcritical in nature, resulting in either  steady or time-dependent asymmetric flows. Furthermore,  we demonstrate that different classes of flows  (steady, periodic, pre-periodic, or quasi-periodic), at times with two-fold multiplicity, emerge as a result of symmetry-breaking bifurcations. Our results also provide new theoretical insights into previous experimental observations in quasi-two-dimensional square vortex flows.