It is shown that within the limits of quasi-geostrophic scaling, biharmonic mixing of potential vorticity is equivalent to biharmonic lateral friction together with biharmonic isopycnal thickness mixing in primitive equations, all with the same viscosity. The problem of unbalanced zonal forces related to the planetary vorticity gradient disappears using a biharmonic instead of a harmonic closure. The former is appropriate for eddy-permitting ocean models which are replacing now more and more the non-eddy-resolving model versions in coupled state-of-the-art climate models. Numerical experiments demonstrate that including biharmonic thickness mixing has some benefits regarding stronger eddy production in marginally eddying ocean models, in particular when the grid size is larger than the baroclinic Rossby radius. Going to finer grids where relative vorticity dominates, however, using only biharmonic friction becomes similar to biharmonic mixing of potential vorticity, justifying the long-standing use of biharmonic friction in reasonably well resolved eddy-permitting ocean models.