A depth-independent isotropic Gent-McWilliams (GM) transport parameter [kappa] is diagnosed from an idealized eddy-resolving primitive equation simulation. The optimal depth-independent isotropic GM parameterization is only able to model less than 50% of the diagnosed total tendency of temperature induced by unresolved mesoscale eddies. A spatio-temporal stochastic model of the GM parameter is developed based on the diagnosed values; the graphical lasso is used to estimate the spatial correlation structure. The stochastic model is used as a stochastic parameterization in low-resolution model simulations. The low-resolution stochastic simulation does a poor job of reproducing the temporal mean of large-scale temperature. Deterministic GM parameterizations and multiplicative stochastic GM parameterizations with unrealistic structure result in significantly more-accurate large-scale temperature in the low-resolution simulations. These results suggest that either the depth-independence or the isotropy of the GM parameterization are unrealistic as models of the eddy tracer transport, or that a stochastic GM parameterization should include an additive component.