  The two-dimensional (2-D) Euler equations of a perfect fluid possess a
beautiful geometric description: they are reduced geodesic equations on the
infinite-dimensional Lie group of symplectomorphims with respect to a
right-invariant Riemannian metric. This structure enables insights to Eulerian
and Lagrangian stability via sectional curvature and Jacobi equations.
  The Zeitlin model is a finite-dimensional analog of the 2-D Euler equations;
the only known discretization that preserves the rich geometric structure.
Theoretical and numerical studies indicate that Zeitlin's model provides
consistent long-time behaviour on large scales, but to which extent it truly
reflects the Euler equations is mainly open. Towards progress, we give here two
results. First, convergence of the sectional curvature in the Euler--Zeitlin
equations on the Lie algebra $\mathfrak{su}(N)$ to that of the Euler equations
on the sphere. Second, $L^2$-convergence of the corresponding Jacobi equations
for Lagrangian and Eulerian stability. The results allow geometric conclusions
about Zeitlin's model to be transferred to Euler's equations and vice versa,
which might be central in the ultimate aim: to characterize the generic
long-time behaviour in perfect 2-D fluids.
