  We consider the hydrodynamics of an incompressible fluid on a 2D periodic
domain. There exists a family of stationary solutions with vorticity given by
$\Omega^*=\alpha\cos (\mathbf{p} \cdot \mathbf{x} )+\beta \sin (\mathbf{p}
\cdot \mathbf{x} )$. This situation can be approximated as a structure
preserving finite dimensional Hamiltonian system by a truncation introduced by
Zeitlin (1990,2005) or by the more standard Galerkin style finite element
method. We use these two truncations to analyse the linear stability of these
solutions and analytical and numerical results are compared. Following the
methods used by Li (2000) the problem is divided into subsystems and we prove
that most subsystems are linearly stable. We derive a sufficient condition for
a subsystem to be linearly unstable and derive an explicit lower bound for the
associated real eigenvalues independent of the truncation size $N$. Then we
show that the corresponding eigenvectors are in $\ell^2$. This together with
known stability results for the 2D periodic Euler equations allows us to
conclude that most of these stationary solutions are nonlinearly unstable. We
confirm our results with a numerical computation of the spectrum for a large,
finite truncation. Finally we discuss the essential spectrum of the full
problem as the limit of the truncated problem.
