In this talk, we will consider the Euler equations describing the flow of an incompressible ideal fluid on the sphere, relevant for geophysical applications. In his pioneering work, V. I. Arnold showed that ideal fluid motions describe geodesics on the Lie group of volume-preserving diffeomorphisms. A finite-dimensional approximation that preserves this geometric structure was proposed by V. Zeitlin. It consists of an isospectral Lie–Poisson flow on the Lie algebra of traceless skew-Hermitian matrices. We will discuss an approximation of Zeitlin’s model based on a time-dependent, low-rank factorization of the vorticity matrix where a basis of eigenvectors is evolved according to the Euler equations. The resulting approximate flow remains isospectral and Lie–Poisson, and the error in the solution, in the approximation of the Hamiltonian and of the Casimir functions depends only on the approximation of the vorticity at the initial time. The computational complexity of solving the approximate model is shown to scale quadratically with the order of the vorticity matrix and linearly when an additional approximation of the stream function is introduced.