On instabilities of the Taylor-Green vortex in 2D Euler flows*

We consider Euler flows on a two-dimensional (2D) periodic domain and are interested in the stability, both linear and nonlinear, of a simple equilibrium given by the 2D Taylor-Green vortex. As the first main result, numerical evidence is provided for the fact that such flows possess unstable eigenvalues embedded in the band of the essential spectrum of the linearized operator, which suggests that passage from linear to non-linear instability in the energy metric is likely for such flows. The latter is confirmed numerically by demonstrating exponential transient growth of the nonlinear dynamics with the rate given by the real part of the unstable eigenvalue. However, the unstable eigenfunction is a distribution, rather than a smooth function, with singularities at the hyperbolic stagnation points of the base flow. As the second main result, we illustrate a fundamentally different, nonmodal, growth mechanism involving a continuous family of uncorrelated functions, instead of an eigenfunction of the linearized operator. Constructed by solving a suitable PDE optimization problem, the resulting flows saturate the known estimates on the growth of the semigroup related to the essential spectrum of the linearized Euler operator as the numerical resolution is refined. These findings are contrasted with the results of earlier studies of a similar problem conducted in a slightly viscous setting where only the modal growth of instabilities was observed. This highlights the special stability properties of equilibria in inviscid flows.