A WENO Finite‐Volume Scheme for the Evolution of Potential Vorticity in Isopycnal Ocean Models

Abstract Ocean circulations are strongly constrained by rotation and stratification, making potential vorticity (PV) a key dynamical property. Specifically, PV is conserved following water parcels in the absence of diabatic forcing and friction—a desirable feature for isopycnal ocean models. Here, we devise a finite‐volume discretization for PV fluxes in the vector‐invariant formulation of shallow‐water equations. The scheme adopts a seventh‐order weighted essentially non‐oscillatory reconstruction of PV on the Arakawa C‐grid while ensuring PV conservation. To prevent PV singularities arising from vanishing layer thickness—typical in isopycnal ocean models—we propose a thickness weighting for the PV reconstruction. The scheme is implemented in the Modular Ocean Model 6 (MOM6) dynamical core and tested through idealized simulations of baroclinic geostrophic turbulence and wind‐driven double gyres. Compared with existing discretizations in MOM6 that conserve energy and/or potential enstrophy, the new scheme not only substantially reduces dispersive errors but also yields a weak potential enstrophy dissipation near grid scales, and thus permits stable time integration without superfluous horizontal viscosity. Additionally, the scheme supports a strong inverse energy cascade and exhibits an elevated eddy kinetic energy. These properties in turn lead to more realistic ocean energetics in eddy‐permitting simulations for which the energy input by surface wind stress is dissipated through bottom drag. In terms of computational efficiency, the scheme leads to notable improvement in energy spectra in eddy‐permitting simulations with a minor increase in total cost by . Results here are promising for improving the simulation of ocean eddies and large‐scale circulations for climate applications.