In this work we consider a finite dimensional approximation for the 2D Euler equations on the sphere $${\mathbb {S}}^2$$
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, proposed by V. Zeitlin, and show their convergence towards a solution to Euler equations with marginals distributed as the enstrophy measure. The method relies on nontrivial computations on the structure constants of $${\mathbb {S}}^2$$
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, that appear to be new. In the last section we discuss the problem of extending our results to Gibbsian measures associated with higher Casimirs.