  The linear stability of a steady state solution of 2D Euler equations of an
ideal fluid is being studied. We give an explicit geometric construction of
approximate eigenfunctions for the linearized Euler operator $L$ in vorticity
form acting on Sobolev spaces on two dimensional torus. We show that each
nonzero Lyapunov-Oseledets exponent for the flow induced by the steady state
contributes a vertical line to the essential spectrum of $L$. Also, we compute
the spectral and growth bounds for the group generated by $L$ via the maximal
Lyapunov-Oseledets exponent. When the flow has arbitrarily long orbits, we show
that the essential spectrum of $L$ on $L_2$ is the imaginary
