Parameterizations of the eddy-induced velocity that advects tracers in addition to the Eulerian mean flow are traditionally expressed as a downgradient Fickian diffusion of either isopycnal layer thickness or large-scale potential vorticity (PV). There is an ongoing debate on which of the two closures is better and how the spatial dependence of the eddy diffusivity should look like. To increase the physical reasoning on which these closures are based, the authors present a systematic assessment of eddy fluxes of thickness and PV and their relation to mean-flow gradients in an isopycnic eddy-resolving model of an idealized double-gyre circulation in a flat bottom, closed basin. The simulated flow features strong nonlinearities, such as tight inertial recirculations, a meandering midlatitude jet, pools of homogenized PV, and regions of weak flow where β/h dominates the PV gradient. It is found that the zonally averaged eddy flux of thickness scales better with the zonally averaged meridional thickness gradient than the eddy flux of PV with the PV gradient. The reason for this is that the two-scale approximation, which is often invoked to derive a balance between the downgradient eddy flux of PV and enstrophy dissipation, does not hold. It is obscured by advection of perturbation enstrophy, which is multisigned and weakly related to mean-flow gradients. On the other hand, forcing by vertical motions, which enters the balance between the downgradient eddy flux of thickness and dissipation in most cases, acts to dissipate thickness variance. It is dominated by the conversion from potential to kinetic energy and the subsequent downgradient transport of thickness. Also, advection of perturbation thickness variance tends to be more single-signed than advection of perturbation enstrophy, forcing the eddy flux of thickness to be more often down the mean gradient. As a result, in the present configuration a downgradient diffusive closure for thickness seems more appropriate to simulate the divergent eddy fluxes than a downgradient diffusive closure for PV, especially in dynamically active regions where the eddy fluxes are large and in regions of nearly uniform PV.