Two-dimensional fluid flow is often approximated in the laboratory with thin electromagnetically forced fluid layers. The faithfulness of such an experimental model must be considered carefully, however, because the physical world is inherently three-dimensional. By adapting an analysis technique developed for oceanographic data, we divide velocity measurements from a thin-layer flow into two components: one that is purely two-dimensional and another that accounts for all out-of-plane flow. We

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... xamine the two-and three-dimensional components separately, finding that motion in thin-layer flows is nearly two-dimensional at low Reynolds numbers, but that out-of-plane flow grows quickly above a critical Reynolds number. This onset is likely due to a shear instability.