Ocean mesoscale eddies contain a large proportion of energy and have an important impact on large-scale circulations. They are found everywhere and are particularly intensive in the western boundary currents such as the Gulf Stream and the Kuroshio. However, these eddies are often not resolved in ocean general circulation models, since the deformation radius in the ocean is at most of the order of 100 km. In particular, the effects of the mesoscale eddies need to be properly parameterized in coarse-resolution ocean models. The most successful parameterization of the mesoscale effect is based on the eddy-induced advection scheme (Gent & McWilliams, 1990; Gent et al., 1995; Griffies, 1998), which mimics the impact of the baroclinic instability by flattening the isopycnal surfaces to transfer the available potential energy (PE) of the resolved flow to the eddy kinetic energy at the sub-grid scales. However, this traditional scheme does not account for the backscattering of kinetic energy (KE) from small to large scales. Some complementary schemes (Bachman, 2019; Bachman et al., 2017; Grooms et al., 2015) have been developed in order to perform more efficient energy transfers for the resolved flow. Classical eddy viscosity models are introduced in coarse models to mimic the mixing action of the unresolved scales. The associated energy dissipation mechanism is often represented by some functional operators (Bachman et al., 2017; Griffies & Hallberg, 2000; Leith, 1971) that depend on the resolved flow. These highly scale-selective, state-dependent friction operators have proven to be very useful for eddy-permitting