Mesoscale eddies play a substantial role in the dynamics of the ocean, but the dominant length-scale of these eddies varies greatly with latitude, stratification and ocean depth. Global numerical ocean models with spatial resolutions ranging from 1° down to just a few kilometers include both regions where the dominant eddy scales are well resolved and regions where the model’s resolution is too coarse for the eddies to form, and hence eddy effects need to be parameterized. However, common parameterizations of eddy effects via a Laplacian diffusion of the height of isopycnal surfaces (a Gent–McWilliams diffusivity) are much more effective at suppressing resolved eddies than in replicating their effects. A variant of the Phillips model of baroclinic instability illustrates how eddy effects might be represented in ocean models. The ratio of the first baroclinic deformation radius to the horizontal grid spacing indicates where an ocean model could explicitly simulate eddy effects; a function of this ratio can be used to specify where eddy effects are parameterized and where they are explicitly modeled. One viable approach is to abruptly disable all the eddy parameterizations once the deformation radius is adequately resolved; at the discontinuity where the parameterization is disabled, isopycnal heights are locally flattened on the one side while eddies grow rapidly off of the enhanced slopes on the other side, such that the total parameterized and eddy fluxes vary continuously at the discontinuity in the diffusivity. This approach should work well with various specifications for the magnitude of the eddy diffusivities.