There are numerous examples of long-lived anticyclones over topographic basins and troughs in the ocean. These are usually surface-intensified and frequently co-exist with a basin-scale cyclonic circulation. The latter is predicted by existing barotropic theory, but the anticyclone is not. Here we consider a variational solution in two layers. The solutions which have minimum enstrophy are bottom-intensified, with cyclonic circulation in a basin. These have negative potential vorticity in both layers, which become homogenized at large energies. The variational problem also yields surface-intensified solutions, with negative potential vorticity at the surface. Whether these have minimum or maximum enstrophy, or whether they are stable, is undetermined. In numerical simulations with a quasi-geostrophic model and random initial conditions, vortex mergers are asymmetric; while cyclones are preferentially sheared out, the anticyclones retain an axisymmetric shape and increase in size from mergers. This often produces a small central anticyclone at depth (or in a single layer), but a more substantial anticyclone at the surface. The latter has a negative q-ψ\psi relation, consistent with the surface-intensified variational solution. Similar results obtain in a full complexity ocean model but with a more pronounced asymmetry in surface vortex mergers and stronger deep vortex flows.