  We consider Euler flows on 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 as well 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, and is
unbounded in the neighbourhood of 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.
